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Simple
Simple
7:02 AM
11
A: Prove the set of sequences $c_0$ which converge to zero in $l_{\infty}$ is closed.

JochenOne more proof: $f:\ell_\infty \to \mathbb R$, $x\mapsto \lim\sup |x_n|$ is continuous so that $c_0= f^{-1}(\lbrace 0\rbrace)$ is closed.

I don't quite understand the accepted solution
 
 
2 hours later…
Martin Sleziak
Martin Sleziak
9:08 AM
@Simple For every continuous map, the preimage of a close set is continuous.
So if we know that $f$ is continuous, the fact that $\{0\}$ is closed in $\mathbb R$ implies that $c_0=f^{-1}(\{0\})$ is continuous.
To check that $f$ is continuous, it is sufficient to notice that $\|f(x)\| \le \|x\|$. Which follows form $\limsup |x_n| = \le \sup |x_n|$.
Sorry, I meant to write closed set rather than close set.
 

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