| Front | Theorem 3.4.5 Divergence Criteria |
| Back | [latex] If a sequence $X = (x_n)$ of real numbers has either of the following properties, then $X$ divergent\begin{enumerate}\item X has two convergent subsequences $X' = (x_{n_k})$ and $X'' = (x_{r_k})$ whose limits are not equal\item $X$ is unbounded\end{enumerate}[/latex] |
| Tags |
| Front | Definition 11.4.3 Neighborhoods of Metric Spaces |
| Back | [latex] Let $(S, d)$ be a metric space. For $\epsilon > 0$, the $\epsilon$-neighborhoood of a point $x_0$ in $S$ is the set $$V_\epsilon(x_0) := \{ x \in S : d (x_0, x) < \epsilon) \}$$[/latex] |
| Tags |
| Front | Theorem 3.4.11 Implications of Limit Superior |
| Back | If (x_n) is a bounded sequence of real numbers, then the following statements for a real number x* are equivalent:[latex] \begin{enumerate}\item $x\mbox{*} = \lim \sup(x_n)$\item If $\epsilon > 0$, there are at most a finite number of $n \in \mathbb{N}$ such that $x\mbox{*} + \epsilon < x_n$, but an infinite number of $n \in \mathbb{N}$ such that $x\mbox{*} - \epsilon < x_n$\item If $u_m = \sup( x_n : n \geq m)$, then $x\mbox{*} = \inf(u_m : m \in \mathbb{N}) = \lim(u_m)$.\item If $S$ is the set of subsequential limits of $(x_n)$, then $x\mbox{*} = \sup S$.\end{enumerate}[/latex] |
| Tags |