Show that the unit ball of Hilbert space is not compact.

Proof.

Let . For each , define by , if , and . Let .

Now, let . Then , where , and .

Now, for each , for some . Then let . Finally, let ,

Then covers . Note that for , . Also, for any , and . Hence for each we need at least one set from , for some . Hence there is no finite subcover.

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This is way too involved, and I do not quite get it. Is it not easier to find a (completely obvious) sequence without any converging subsequence?

Oh, I see. , where if and otherwise. Actually, my proof above was based on that sequence originally, but I didn’t think about the convergent subsequence characterization of compactness. Oops, lol.