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\begin{document}
\title{Infi - Finals}
\author[1]{Ricva E Hildebrandt}%
\affil[1]{The Hebrew University of Jerusalem}%
\vspace{-1em}
\date{\today}
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\sloppy
\section*{Week 3 - Lecture 7}
\section*{Minima and maxima of a set}
Let A be a non-empty subset of the real numbers. Let \(M,\ m\ \in A\). Let's suppose that:
\begin{itemize}
\item \(M\ =\ \max\left(A\right)\ \Leftrightarrow M\ \in A\ \wedge\ M\ =\ \sup (A)\)
\item \(m\ =\ \min\ \left(A\right)\ \Leftrightarrow m\ \in A\ \wedge\ \text{m}=\ \inf\left(A\right)\)
\end{itemize}\\
\subsection*{Corrolary on the how two numbers are equal}
Let \(x\ \in\mathbb{R}\) and \(m,\ n\ \in\mathbb{Z}\). The numbers m and n are said to be equal if: \\
\(x-1\ <\ n\ \le x\) and \(x-1\ <\ m\ \le x\)
In other words, two integers cannot be in between a real number and its successor, unless they are the same number.\\
\subsection*{On a upper bounded set}
Let \(b\ \in\mathbb{R}\). Then, the set \(A_b\ =\left\{n\ \in\mathbb{Z}\ ,\ n\le b\right\}\).
In other words, b is a real number and \(A_b\) is the set of all integers smaller than or equal to b. Then, \(A_b\) is bounded from above.\\\
\section*{Floor and ceiling functions}
These two functions make it possible to "round" the value of a number.
\subsection*{Integral part or integer part}
Let b be a real number. The value sup(\(A_b\)) will be called the integer part of b and will be represented by: \(\lfloor b\rfloor\).\\
\section*{Theorem - b in relation to its integer part}
The number between \(\lfloor b\rfloor\ \le b\ <\ \lfloor b\rfloor+1\) is b itself and \(\lfloor b\rfloor\) is an integer number.\\
\subsection*{Conclusion derived from the previous theorem}
It follows from the previous that in every interval greater than one contains an integer.
Let \(x,\ y\ \in\mathbb{R}\). If \(y\ >\ x+1\), then:\\
\(\exists\ n\ \in\mathbb{Z}\), such as:\\
\(x\ \ 0\ \wedge\ x^2\ \le2\right\}\)\\
\(B\ =\ \ \left\{x\ \ in\ \mathbb {F} :\ x\ >\ 0\ \wedge\ x^{2\ }\ \ge\ 2\right\}\)\\
If there was \(c\ \in\mathbb {F} \), such that:\\
\(\forall\ a\ \in A,\ \forall\ b\ \in B,\ a\le c\le b\), then \(c^2\ =\ 2\)\\
\section {Irrational numbers}
A real number is called irrational if and only if: \(x\ \in\mathbb{R},\ x\ \notin\mathbb{Q}\)
\section*{n-ordered roots}
For any \(n\ \in\mathbb{N}\), let's define \(a^n\). And for all real number \(a\):\\
\begin{itemize}
\item \(a^1\ =\ a\)
\item \(a^{n+1}\ =\ a^n\cdot a\)
\end{itemize}\\
\section*{Theorem - the existence of the nth root of a real number}
For all positive real number and for all natural number, there is another real number, such that the first real number is the root of the latter. in mathematical notation:\\
\(\forall a\ \in\mathbb{R},\ \forall n\ \in\mathbb{N},\ \exists x\ \in\mathbb{R},\ x^n\ =\ a\)\\
\section*{Power of a number}
\subsection*{Definition}
Let \(a\) be a positive number. And let \(r\)be any rational number, which can be written as: \(r\ =\ \frac{m}{n}\), such that \(m\ \in\mathbb{Z}\) and \(n\ \in\mathbb{N}\). We define \(a^r\) as:\\
\(a^r\ =\ a^{\frac{m}{m}}\ =\ (\sqrt[n]{a})^m\)\\
\section*{Real functions of a real variable}
\subsection*{Definition of a function}
Let there be two sets A and B, which are subsets of \(\mathbb{R}\). A function is defined by any law that connects every \(a\ \in A\) to an element \(b\ \in B\). The mathematical notation used is: \(f:\ A\ \rightarrow\ B\).
\subsection*{Injective function}
\subsubsection*{Definition}
A function \(f:\ A\ \rightarrow\ B\) will be called injective if and only if:\\
\(\forall a_1,\ a_2\ \in A,\ f\left(a_1\right)\ =\ f\left(a_2\right)\ \implies a_1\ =\ a_2\)\\
This is equivalent to:\\
\(\forall a_1,\ a_2\ \in A,\ a_1\ne a_2\ \Rightarrow f\left(a_1\right)\ \ne f\left(a_2\right)\)\\
\subsection*{Surjective function}
\subsubsection*{Definition}
A function \(f:\ A\ \rightarrow\ B\) is surjective if and only if:\\
\(f\left(A\right)\ =\ Im\left(f\right)\ =\ B\)\\
or:\\
\(\forall b\ \in B\ \exists a\ \in A,\ f\left(a\right)\ =\ b\)\\
\subsection*{Bijective function}
\subsubsection*{Definition}
A function \(f:\ A\ \rightarrow\ B\) is said to be bijective if it is injective and surjective at the same time. In other words, \\
\(\forall b\ \in B\ \exists!a\ \in A,\ f\left(a\right)\ =\ b\)
\subsection*{Inverse function}
Let \(f\) be a function such that: \(f:\ A\rightarrow\ B\), its inverse will be \(g:\ B\ \rightarrow\ A\).\\
\subsection*{Important characteristics of a surjective function}
If \(f\) is surjective, then:\\
\begin{itemize}
\item \(f^{-1}\), i.e. its inverse, is also surjective
\item \(\forall a\ \in A,\ f^{-1}\left(f\left(a\right)\right)\ =\ a\)
\item \(\forall b\ \in B,\ f^{-1}\left(f\left(b\right)\right)\ =\ b\)
\item \(\left(f^{-1}\right)^{-1}\ =\ f\)
\end{itemize}
\section*{Limit of a real function}
\subsection*{Point limit}
\subsubsection*{Definition}
Let \(x_0\)be any real number. Let's define two different typse of neighborhood.
\begin{enumerate}
\item Deleted neighborhood \\
\item Complete neighborhood\\
\end{enumerate}
\subsection*{Complete neighborhood}
\(\left(x_0\ -\ \delta,\ x_0\ +\ \delta\right)\), such that \(0\ <\ \delta\ \in\mathbb{R}\). \\
In other words, a complete niehgborhood is a subset that can be defined as:\\
\(\left(x_0\ +\ \delta,\ x_0\ -\ \delta\right)\ =\ \) \(\left\{x\ \in\mathbb{R},\ \right|x-x_0\left|<\delta\right\}\ =\ \left\{x\ \in\mathbb{R},\ dist\left(x,\ x_0\right)<\delta\right\}\)\\
It means that for every given real number in this neighborhood, the distance between this number and \(x_0\) is very small, i. e. small than \(\delta\).
\subsection*{Deleted neighborhood}
For every given real number in this neighborhood, the distance between this number and \(x_0\) is somewhere between 0 and \(\delta\). Or:\\
\(\left(x_0-\delta,\ x_0+\delta\right)\ =\ \left\{x\in\mathbb{R},\ 0<\left|x-x_0\right|<\delta\right\}\ =\ \left\{x\in\mathbb{R},\ 00,\ \exists\delta>0,\ \forall x\in D,\ 0<\left|x-x_0\right|<\delta\ \Rightarrow0,\left|f\left(x\right)-L\right|<\epsilon\)
\section*{Week 5}
\section*{Lecture 10}
\section*{Direct conclusions from the definition of limits}
\subsection*{Upper bounded set}
Let \(f:D\ \rightarrow\ \mathbb{R}\) be a function well defined in the deleted neighborhood of \(x_0\). suppose that the limit \(\lim_{x\rightarrow\mathbb{R}}f\left(x\right)\ =\ L\in\mathbb{R}\) exists. There is a neighborhood \(U\) of \(x_0\) in which the function is limited from above. It is easier to think about it as \(U\) being a upper-bounded set. In mathematical notation, \\
\(\left\{f\left(U\right)\ =\ \left\{f\left(x\right)\right|x\in U\right\}\)
\subsection*{Corollary on the sign of L}
Let \(f:D\rightarrow\mathbb{R}\) be a function defined in a deleted neighborhood of \(x_0\) and the limit \(\lim_{x\rightarrow x_0}f\) exists. Then, if:\\
\begin{enumerate}
\item \(L>0:\ \frac{L}{2}0,\ \exists\delta>0,\ \forall x\in\mathbb{R},\ 0M\)//
The same definition can be expanded for minus infinity.
\subsubsection*{Important theorems on limits at the infinity}
Let \(f\) and \(g\) be two real functions, defined on the same domain and also defined at \(x_0\). Suppose that \(\lim_{x\rightarrow x_0}f\ =\ \infty\). Then,//
\begin{enumerate}
\item The function \(f\) is bounded from below, but not from above.
\item Let \(f\le g\). Then, \(\lim_{x\rightarrow x_0}g\ =\ \infty\). In other words, if \(g\) is always above \(f\), then \(g\) is not bounded from above.
\item If you multiply the function by -1, its limit will tend to minus infinity.
\item If \(g\) is bounded from below, and its limit exists and is either finite or not. Then, the limit of the sum of both functions also tends to infinity. Mathematically, \(\lim_{x\rightarrow x_0}\left(f+g\right)\ =\ \infty\)
\item Let \(c>0\) and \(g>c\), then the product of both functions tend to infinity, such that: \(\lim_{x\rightarrow x_0}f\cdot g\ =\ \infty\)
\item Let \(c<0\) and \(g0\ \ \exists N\in\mathbb{R}\ \forall x\in\mathbb{R}\)//
\(x>N\ \Rightarrow\left|f\left(x\right)\ -L\right|<\epsilon\)//
If the limit exists, it is unique.
\subsubsection*{Theorem on the limits of two real functions}
Let there be two real functions \(f\) and \(g\), whose limits tend to the infinity exist and are finite. Then,//
\begin{itemize}
\item There is an open interval, not bounded from above, i.e. \(\left(a,\ \infty\right)\), in which \(f\) is bounded.
\item If \(f\le g\), then \(L\le M\). Being that \(\lim_{x\rightarrow\infty}f\ =\ L\) and \(\lim_{x\rightarrow\infty}g\ =\ M\)
\item Similarly, if \(Mf\left(x_0\right)\) in a right-sided complete vicinity of \(x_0\).
\subsubsection*{Corollary on the relation between the derivative and the direction of the function}
If the derivative is positive at \(x_0\), then the function is increasing at that point. If the derivative is negative, then the function is decreasing there.
\subsection*{Local maximum}
\subsubsection*{Definition}
There is a local maximum to the function \(f\) if \(f\left(x\right)\le f\left(x_0\right)\).
\subsection*{Local minimum}
\subsubsection*{Definition}
There is a local minimum to the function \(f\) if \(f\left(x\right)\ge f\left(x_0\right)\).
\subsubsection*{Fermat's theorem: relation between derivative and critical points}
Let \(f\) be a real function defined on a deleted vicinity of \(x_0\). If there is a critical point for \(f\) at \(x_0\). Then, the derivative of the function at that point is zero.
\subsubsection*{Rolle's theorem: existence of critical point when the edges of the function are equal}
Let \(f\) be a real function defined on a domain which is a subset of \(\mathbb{R}\). If \(f\) is differentiable in its domain, it is continuous in its domain and it is differentiable at any internal point of its domain. Suppose that the function \(f\) is defined as: \(f:\left[a,\ b\right]\rightarrow\mathbb{R}\). Suppose that the image of the edges of the domain are equal, i.e. \(f\left(a\right)=f\left(b\right)\). Then, there is a point inside the domain, which is not its edges, for which the function has a critical point. In other words, \(\exists x\ \in\ \left(a,\ b\right),\ f'\left(x_0\right)\ =\ 0\).
\subsubsection*{Lagrange's theorem: derivative as the slope of the function}
Let there \(f\) be a real function defined as \(f:\left[a,b\right]\rightarrow\mathbb{R}\). There is an internal point at the domain, for which the derivative of the function is its slope. The conclusion of this theorem is that if the derivative is always zero for all internal points in the domain, then the function is constant.
\subsubsection*{Conclusions based on the previous theorem}
Let there be two real functions \(f\) and \(g\), defined on the same domain. If both functions have the same derivative for all points of their domain, there is a real number \(c\), such that \(f\ =\ g+c\).
If the derivative is positive, then the function is increasing in the closed interval of the domain. The same reasoning can be used if the derivative is negative.
\subsection*{Higher order derivatives}
\subsubsection*{Theorem on the second derivative}
Let \(f\) be a real function that can be derived twice. Suppose that \(x_0\) is a critical point of \(f\). Then,//
\begin{enumerate}
\item If the second derivative at that point is negative, then the function is decreasing there.
\item Something similar can be said if the second derivative is positive, then the function is increasing there.
\item If the second derivative is equal to zero and \(x_0\) is a local maximum, then the function is decreasing there.
\item If the second derivative is equal to zero and \(x_0\) is a local minimum, then the function is increasing at a vicinity of that point.
\end{enumerate}
\section*{Week 10}
\section*{Lecture 23}
\subsection*{Darboux sums}
\subsubsection*{Definition of partition}
Let there \(a\) and \(b\) be two real numbers, such that \(a < b\). The set \(P\) is defined as a partition of the interval \(\left[a,\ b\right]\) when \(P\) can be written as: \(P\ =\ \left\{x_0,\ x_1,\ ...,\ x_n\right\}\), such that: \(a\ =\ x_0\ <\ x_1\ <\ ...\ <\ x_n\ =\ b\). In other words, \(P\) is a set of points from the interval \([a, b]\), which include \(a\) and \(b\).
\subsubsection*{Definition of the parameter of a partition}
It is the maximum distance between two consecutive points in the partition. It is defined as: \(\Delta\left(P\right)\ =\ \max_{1\le i\le n}\left(x_i\ -\ x_{i-1}\right)\)
\subsubsection*{Definition of Refinement}
Let there be two sets \(P\) and \(Q\), which are both partitions of the interval \(\left[a,\ b\right]\). If \(Q\) is a subset of \(P\), then it is called a refinement of \(P\). The union of both sets is defined as the common refinement of the interval \(\left[a,\ b\right]\).
\subsubsection*{Definition of Lower Darboux Sum}
Let there \(f\) be a real function, whose domain is the interval \(\left[a,\ b\right]\). Let the function be bounded on the domain. And let there \(P\) be a partition of the domain. The lower Darboux sum is defined as: \(L\left(f,\ P\right)\ =\ \sum_{i=1}^nm_i\left(x_i-x_{i-1}\right)\). Being \(m_i\) the infimum of the function on a certain point. In plain English, the lower Darboux sum \(L\) of a function \(f\) according to a partition \(P\) is the multiplication of the infimum of the function on a point by the distance between a point and the point just before that, according to the partition.
\subsubsection*{Definition of Upper Darboux Sum}
Considering true all the information from above, the definition of upper Darboux sum is: \(U\left(f,\ P\right)\ =\ \sum_{i=1}^nM_i\left(x_i\ -\ x_{i-1}\right)\). In other words, the upper Darboux sum \(L\) of a function \(f\) according to a partition \(P\) is the multiplication of the supremum of the function on a point by the distance between a point and the point just before that, according to the partition.
\subsubsection*{Corollary on the relation between lower and upper Darboux sums}
To any partition \(P\) of the domain, the lower Darboux sum will always be smaller or equal than the upper one. or: \(L\left(f,\ P\right)\le U\left(f,\ P\right)\).
\subsubsection*{Corollary on the relation between the refinement of a partition and Darboux sums}
Any refinements of the partition \(P\) will cause the lower sum of the refinement to be greater than the lower sum of \(P\) itself. And any refinement of the partition will cause the upper sum of the refinement to be lower or greater than the sum of the original partition. Being \(\overline{P}\) be the refinement of the partition \(P\), then: \(L\left(f,\ P\right)\le L\left(f,\ \overline{P}\right)\le U\left(f,\ \overline{P}\right)\le U\left(f,\ P\right)\).
\subsubsection*{Conclusions on the relation between infimum, supremum and Darboux sums}
Let the infimum of the function \(f\) be \(m\ =\ \inf_{x\in\left[a,\ b\right]}f\left(x\right)\) and the supremum of the function be \(M\ =\ \sup_{x\in\left[a,\ b\right]}f\left(x\right)\). Any lower and upper Darboux sums of the function according to a partition \(P\) will always be in between the infimum of the function on the interval \(\left[a,\ b\right]\) and the supremum of the function in the same interval. In mathematical notation: \(m\left(b-a\right)\le L\left(f,\ P\right)\le U\left(f,\ P\right)\le M\left(b-a\right)\).
\subsection*{Integral}
\subsubsection*{Definition of integral}
Let \(f\) be a real function, whose domain is the interval \(\left[a\ b\right]\). Let this function be bounded on the domain. Let \(L\) and \(U\) be the lower and upper Darboux sums, respectively. The supremum of the lower sum is said to be the lower integral of \(f\) in the interval \(\left[a,\ b\right]\) and can be written as: \(\int_{\overline{a}}^bf\left(x\right)dx\). The infimum of the upper Darboux sum is said to be the upper integral of \(f\) on the interval \(\left[a,\ b\right]\) and can be written as: \(\int_a^{\overline{b}}f\left(x\right)dx\). A function is said to be integrable if and only if the lower and the upper integral have the same value.In such case, the integral of the function \(f\) over the interval \(\left[a,\ b\right]\) is: \(\int_a^bf\left(x\right)dx\). If the function is positive or zero for any point of the domain, then the integral is said to be the area under the curve.
\section*{Lecture 24}
\subsection*{Integral}
\subsubsection*{Lemma on the slices of a Darboux sum}
Let there be \(f\) a real function, whose domain is the interval \(\left[a,\ b\right]\). Let there be two partitions of the domain \(P_1\) and \(P_2\). Suppose the function is bounded, then:
\begin{itemize}
\item \(f\) is integrable over the domain
\item There is a real number which is: \(L\left(f,\ P_1\right)\le c\ \le U\left(f,\ P_2\right)\)
\item \(\forall\epsilon>0,\ U\left(f,\ P_2\right)-L\left(f,\ P_1\right)<\epsilon\)
\end{itemize}
\subsubsection*{Darboux condition on integrability}
let the function \(f:\ \left[a,\ b\right]\rightarrow\mathbb{R}\) be bounded on the domain. It will be said to be integrable if and only if \(U\left(f,\ P\right)-L\left(f,\ P\right)\ <\ \epsilon\). In other words, the difference between the upper and the lower Darboux sums cannot be zero.
\subsection*{Arithmetics of Integrals}
\subsubsection*{Theorem on addition of integrals and multiplication of an integral by a scalar}
Let \(f\) and \(g\) be two real function, whose domain is the interval is \(\left[a,\ b\right]\). Then,
\begin{itemize}
\item Addition: \(\int_a^bf\left(x\right)dx\ +\ \int_a^bg\left(x\right)dx\ =\ \int_a^bf\left(x\right)+g\left(x\right)dx\)
\item Let there be any real number. \(\int_a^b\lambda\cdot f\left(x\right)dx\ =\ \lambda\int_a^bf\left(x\right)dx\)
\end{itemize}
\subsubsection*{Characteristics of order in integrals}
Let \(f\) and \(g\) be two real function, whose domain is the interval is \(\left[a,\ b\right]\). Then,
\begin{itemize}
\item If \(f\) is positive or zero, its integral over the interval \(\left[a,\ b\right]\) is also positive or zero.
\item If \(f\le g\), then \(\int_a^bf\left(x\right)dx\le\int_a^bg\left(x\right)dx\). Comment: if \(f\) is smaller or equal to \(g\), then their integrals will have the same relation.
\item the functions \(\max\left\{f,\ 0\right\}\) and \(min\left\{f,\ 0\right\}\) are integrable
\item If the absolute value of the function is integrable, then \(\left|\int_a^bf\left(x\right)dx\right|\le\int_a^b\left|f\left(x\right)\right|dx\)
\end{itemize}
\section*{Lecture 25}
\subsection*{Frequency and Integrals}
\subsubsection*{Definition}
Let there be a real function \(f\) bounded on a subset \(I\) of its domain. The frequency of this function on this interval is://
\(\omega\left(f,\ I\right)\ =\ \sup\left\{f\left(x\right)-f\left(y\right):\ x,\ y\in I\right\}\ =\ \sup_{x\in I}f\left(x\right)-\inf_{y\in I}f\left(y\right)\)//
The frequency can also be calculated in relation to a partition \(P\). The maximum frequency \(\omega_{\max}\) is the supremum of all the frequencies on that partition.
\subsubsection*{Theorem on the frequency of a function on a partition}
The frequency will always be positive and non-zero.
\subsection*{Integrability of continuous functions}
Let \(I\) and \(J\) be subsets of the domain, being \(J\) a subset of \(I\). Then, the frequency of the function in \(J\) will always be smaller or equal than the frequency in \(I\). Or: \(\omega\left(f,\ J\right)\le\omega\left(f,\ I\right)\)
\subsubsection*{Theorem on the relation between continuity and integrability}
Continity implies integrability. But the other way around is not true. A function can have certain discontinuities and still be integrable. For it to be happen. The function will still be integrable even if it has a finite number of discontinuities of the first or the second type, which means that either one of the sided-limits do not exist or they are not equal.
\section*{Week 11}
\section*{Lecture 26}
\subsection*{Domain of integration}
\subsubsection*{Theorem on integrability of a function over an interval}
Let there be three real numbers \(a,\ b,\ c\), such that \(a**0\ \exists N\in\mathbb{N}\ \forall n\in\mathbb{N},\ n>N\ \Rightarrow\left|a_n-L\right|<\epsilon\)
if the limit exists, it is unique.
\section*{Lecture 32}
\subsection*{Convergence}
If a series converges it is somehow bounded, it can be either bounded from above, from below or both.
Let there be two series \(a_n\) and \(b_n\). If they converge to \(A\) and \(B\), respectively. Then, it is possible to draw the following conclusions:
\begin{enumerate}
\item If A****0\), for every \(n\).
\subsection*{Corollary on the positiveness and convergence of a series}
Let the sum \(\sum_{n=1}^{\infty}a_n\) is positive and has partial sums \(S_N\). It will converge if and only if the series is bounded and \(\sum_{n=1}^{\infty}a_n\) is the supremum of its partial sums.
\subsection*{Comparison Test}
Let \(\sum_{n=1}^{\infty}a_n\) be positive and convergent. And let there be \(\sum_{n=1}^{\infty}b_n\) be positive. If \(0\le b_n\le a_n\), then \(\sum_{n=1}^{\infty}b_n\) also converges.
\subsection*{Absolute convergence}
\subsubsection*{Definition}
The sequence \(\sum_{n=1}^{\infty}a_n\) is said to converge absolutely if and only if \(\sum_{n=1}^{\infty}a_n\) also converges. If the sequence converges absolutely, then it converges.
\section*{Lecture 35}
\subsection*{Sequences}
\subsubsection*{Special cases}
The sequence \(\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n+1}}{n}\) does not converge absolutely, but it does converge.
If a sequence is reached by another sequence, then if one converges, the other one also converges.
\subsubsection*{New ordering}
If the series \(b_k\) is defined as a new ordering of the series \(a_n\). Then, \(b_k\) converges, because it is bounded by \(a_n\) and \(\sum_{k=1}^{\infty}b_k\ =\ \sum_{n=1}^{\infty}a_n\).
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