Question

Real Analysis help please.

Answer 1

Consider \(\textbf{R}^n\) equipped with \(|| \ ||_2\).
Prove \(E\subset \textbf{R}^n\) is totally bounded iff it is bounded.

Answer 2

Proof:
Let \(E\subset \textbf{R}^n\), assume \(E\) is totally bounded. Then \(E=\large {\cup}_{i=1}^{n}B(x_i,\epsilon)\).
So \(E\subset B(x_1,max\{||x_1||,||x_2||,....,||x_n||\}+\epsilon)\)
Thus E is bounded.

Suppose E is bounded, Then \(\overline{E}\) is closed and bounded, and thus compact. So \(E\subset \overline{E}\) is tottaly bounded.\(_\square\)

Answer 3

I really want confirmation that this works...I am concerned about the very last line.