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Diffstat (limited to 'mindmap/limit.org')
| -rw-r--r-- | mindmap/limit.org | 16 |
1 files changed, 14 insertions, 2 deletions
diff --git a/mindmap/limit.org b/mindmap/limit.org index 65ae15f..22b3280 100644 --- a/mindmap/limit.org +++ b/mindmap/limit.org @@ -49,10 +49,22 @@ We know: \begin{align*} \lim s_{n} = s \iff \forall \epsilon > 0, \exists N, n > N \implies | s_{n} - s | < \epsilon \\ \end{align*} - +which is equivalent to: +\begin{align*} +\lim s_{n} = s \iff \forall \epsilon > 0, \exists N, n > N \implies s - \epsilon < s_{n} < s + \epsilon +\end{align*} and our sequence $\{s_{n}\}$ is monotone. If $\{s_{n}\}$ is increasing, we have: \begin{align*} s_{n + 1} \ge s_{n} \end{align*} -for all n. +for all n. Without loss of generality we shall assume $\{s_{n}\}$ is increasing. Then we take two cases: +1. $\{s_{n}\}$ is bounded. +2. $\{s_{n}\}$ is unbounded. +In the case $\{s_{n}\}$ is bounded: +\begin{align} +\label{} +\exists M, \forall n, s_{n} \le M \\ +s_{0} \le ... \le s_{n} \le s_{n + 1} \le s_{n + 2} \le ... \le M +\end{align} + #+end_proof |