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## Homework Statement

Want to prove that [0,1] in R is compact. Let [itex]\bigcup_{\alpha\in A}[/itex] I[itex]_{\alpha}[/itex] be an open cover of [0,1].

By open sets in R.

Let E={t[itex]\in[/itex][0,1] s.t. [0,t] is covered by a finite number of the open cover sets I[itex]_{\alpha}[/itex]}.

Prove that E[itex]\neq[/itex][itex]\emptyset[/itex].

## The Attempt at a Solution

Let t=0, the set E=[0,0] has only one element, it is non-empty.

Is this ok for the non-empty part?