deletions | additions       diff --git a/begin_claim_The_algorithm_runs__.tex b/begin_claim_The_algorithm_runs__.tex  index b25a48f..1f5c369 100644  --- a/begin_claim_The_algorithm_runs__.tex  +++ b/begin_claim_The_algorithm_runs__.tex 
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&= O\left(n\log n\right) + n^{\log_2 2}O\left(1\right) + \sum_{j=0}^{\log_2 n-1} 2^j O\left(\log n - \log{2^j}\right)\\   &= O\left(n\log n\right) + O\left(n\right) + \left(O\left(\log n\right)\sum_{j=0}^{\log_2 n-1} 2^j - \sum_{j=0}^{\log_2 n-1} O\left(2^j\log{2^j} \right)\right)\\   &=O\left(n\log n\right) + O\left(n\right) + \left(O\left(\log n\right) 2^{\log_2 n} - C\right) \tag{with $C\geq0$}\\   &= O\left(n\log n\right) + O\left(n\right) +  O\left(n\log n\right) - C\\ &\leq &\geq  O\left(n\log n\right) + O\left(n\right) +  O\left(n\log n\right)\\ &\in O\left(n\log n\right)  \end{align*}