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Diffstat (limited to 'mindmap')
| -rw-r--r-- | mindmap/limit.org | 22 | ||||
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| -rw-r--r-- | mindmap/ltximg/org-ltximg_affd5e1b334acdfe2a68107c61ebef5e019d4004.png | bin | 0 -> 322 bytes |
5 files changed, 16 insertions, 6 deletions
diff --git a/mindmap/limit.org b/mindmap/limit.org index 946e4d2..65ae15f 100644 --- a/mindmap/limit.org +++ b/mindmap/limit.org @@ -28,21 +28,31 @@ ordering: \forall a \in \mathbb{R}, - \infty < a < +\infty \end{align*} defined. Note that we can define equivalence relations on these symbols, but algebra reamins undefined. +** Unbounded Sequences +Unbounded sequences can still limit to $+\infty$ or $-\infty$, although the limit does not exist +for many unbounded sequences. If a sequence is one of: +1. unbounded above +2. unbounded below +but not both, it is possible that such sequences limit to $\infty$. ** Limits on Monotone Sequences An increasing sequence is a sequence $\{s_{n}\}$ defined such that: \begin{align*} \forall n \in \mathbb{N}, \forall m \in \mathbb{N}, n \ge m \implies s_{n} \ge s_{m}. \end{align*} -and now we wish to prove that the limit of monotone sequences always exist. - -\begin{align*} -\lim s_{n} = s \iff \forall \epsilon > 0, \exists N, n > N \implies | s_{n} - s | < \epsilon \\ -\end{align*} #+begin_theorem -If I am bad, then you are too. +The limit of monotone sequences always exists. #+end_theorem #+begin_proof +We know: +\begin{align*} +\lim s_{n} = s \iff \forall \epsilon > 0, \exists N, n > N \implies | 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. #+end_proof diff --git a/mindmap/ltximg/org-ltximg_11c54a0d8c448cacab472b239324ff746f460d75.png b/mindmap/ltximg/org-ltximg_11c54a0d8c448cacab472b239324ff746f460d75.png Binary files differnew file mode 100644 index 0000000..d16e722 --- /dev/null +++ b/mindmap/ltximg/org-ltximg_11c54a0d8c448cacab472b239324ff746f460d75.png diff --git a/mindmap/ltximg/org-ltximg_425282e0ea5b96b0a0be0d172dc5f75204b89dc0.png b/mindmap/ltximg/org-ltximg_425282e0ea5b96b0a0be0d172dc5f75204b89dc0.png Binary files differnew file mode 100644 index 0000000..f44612e --- /dev/null +++ b/mindmap/ltximg/org-ltximg_425282e0ea5b96b0a0be0d172dc5f75204b89dc0.png diff --git a/mindmap/ltximg/org-ltximg_4f4d02f8e191ed1e27957d9377b37357d3fa0f82.png b/mindmap/ltximg/org-ltximg_4f4d02f8e191ed1e27957d9377b37357d3fa0f82.png Binary files differnew file mode 100644 index 0000000..1a9c609 --- /dev/null +++ b/mindmap/ltximg/org-ltximg_4f4d02f8e191ed1e27957d9377b37357d3fa0f82.png diff --git a/mindmap/ltximg/org-ltximg_affd5e1b334acdfe2a68107c61ebef5e019d4004.png b/mindmap/ltximg/org-ltximg_affd5e1b334acdfe2a68107c61ebef5e019d4004.png Binary files differnew file mode 100644 index 0000000..04ca7f1 --- /dev/null +++ b/mindmap/ltximg/org-ltximg_affd5e1b334acdfe2a68107c61ebef5e019d4004.png |
