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Show that the sequence $x(n)=\frac{1}{n}+\frac{1}{n+1}+\ldots +\frac{1}{2n}$ is monotone decreasing and bounded below and therefore convergent.   x(n) is monotone decreasing if $x(n+1)\leq x(n)$ for all $n\geq 1$. \[ x(n+1) = \frac{1}{n+1}+\frac{1}{n+2}+\ldots +\frac{1}{2n+1} \]