Let be a compact metric space and let be a sequence of compact sets in . Is it true that is compact?

**Solution**

The answer is no since we can take and . Hence

which is clearly not compact.

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# Is the infinite union compact?

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Let be a compact metric space and let be a sequence of compact sets in . Is it true that is compact?

**Solution**

The answer is no since we can take and . Hence

which is clearly not compact.