🍈 Zettelkasten

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Apr 19, 20251 min read

Proof

Suppose $\sum a_{n}$ converges to $s \in R$
Then, by the definition of convergence, $lim_{n \to \infty} S_{n} = s$
WTS: $lim_{n \to \infty} a_{n} = 0$
Then, consider $lim_{n \to \infty} a_{n} = lim_{n \to \infty} (a_{n} + 0)$
$= lim_{n \to \infty} (a_{n} + a_{1} + \dots + a_{n - 1} - a_{1} - \dots - a_{n - 1})$
$= lim_{n \to \infty} (S_{n} - S_{n - 1})$ by Series Partial Sum
$= lim_{n \to \infty} (S_{n}) - lim_{n \to \infty} (S_{n - 1})$ by Limit Properties
$= lim_{n \to \infty} (S_{n}) - lim_{n \to \infty} (S_{n})$ as $lim_{n \to \infty} (n - 1) \sim n$
$= s - s$
$= 0$
$□$