## Proof of the Week for 20120813

Cauchy convergence criterion for a series of real numbers: The series $\sum_{n=1}^\infty a_n$ converges if and only if for $\epsilon > 0$, there exists $N \in \mathbb{N}$ such that whenever $n, m \ge N$, $\left|a_{m+1} + a_{m+2} + \cdots + a_n\right| < \epsilon$.

Proof: Let $\left(S_n\right)$ and $\left(S_m\right)$ be sequences of partial sums of the series $\sum a_n$, such that $S_n = \sum_{p=1}^n a_p$, $S_m = \sum_{q=1}^m a_q$, where $n > m$. Invoking the Cauchy convergence criterion for sequences, we have $\left|S_n - S_m\right| < \epsilon$. $\left|S_n - S_m\right| = \left|a_{m+1} + a_{m+2} + \cdots + a_n\right| < \epsilon$.

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