Fix typos introduces more
This commit is contained in:
Zheyuan Wu
2024-12-03 11:20:59 -06:00
parent cbed1333ed
commit 9283c6b427
21 changed files with 213 additions and 44 deletions

View File

@@ -2,9 +2,9 @@
## Relations between series and topology (compactness, closure, etc.)
Limit points $E'=\{x\in\mathbb{R}:\forall r>0, B_r(x)\backslash\{x\}\cap E\neq\emptyset\}$
Limit points $E'=\{x\in\mathbb{R}:\forall r>0, B_r(x)\backslash\{x\}\cap E\neq\phi\}$
Closure $\overline{E}=E\cup E'=\{x\in\mathbb{R}:\forall r>0, B_r(x)\cap E\neq\emptyset\}$
Closure $\overline{E}=E\cup E'=\{x\in\mathbb{R}:\forall r>0, B_r(x)\cap E\neq\phi\}$
$p_n\to p\implies \forall \epsilon>0, \exists N$ such that $\forall n\geq N, p_n\in B_\epsilon(p)$
@@ -24,7 +24,7 @@ Rudin Proof:
Rudin's proof uses a fact from Chapter 2.
If $E$ is compact, and $S\subseteq E$ is infinite, then $S$ has a limit point in $E$ ($S'\cap E\neq\emptyset$).
If $E$ is compact, and $S\subseteq E$ is infinite, then $S$ has a limit point in $E$ ($S'\cap E\neq\phi$).
## Examples of Cauchy sequence that does not converge