Professors Road Map to Glory.
- Explain why if $\{a_n\}_{n}^{\infty}$ is a sequence of numbers, and $$ a_n \in E_1\cup E_2,...\cup E_n$$ for all $n\in \mathbb{N}$, then their is a subsequence $\{a_{n(k)}\}_{k}^{\infty}$ completely contained in one of the compact sets $E_k$ for some $k \in \{1,2,...n\}$
- Explain why, for the subsequence $\{a_{n(k)}\}_{k}^{\infty}$ contained completely in one compact set $E_k$, there is a convergent subsequence of the subsequence with a limit contained in $E_k$.
these are current defenitions that I have for compactness, looking at the RTG, I more than likely will use #2
A set $E$ is compact iff
- Every open cover of E permits a finite subcover.
- Every sequence of values in E contains a convergent subsequence whose limit is in E.
- E is closed and bounded.
My intuition:
If $\{a_n\}_{n}^{\infty}$ is a sequence in a union of compact sets.
I am super hazy in the part that reads, "then their is a subsequence $\{a_{n(k)}\}_{k}^{\infty}$ completely contained in one of the compact sets $E_k$ for some $k \in \{1,2,...n\}$"
however given then there is a subsequence $\{a_{n(k)}\}_{k}^{\infty}$ completely contained in one of the compact sets $E_k$ for some $k \in \{1,2,...n\} $ the subsequence of the subsequence is convergent to a point in $E_k$ thus by the second definition of compactness, the union of the sets are compact.
