Is the infinite union compact?

Let M be a compact metric space and let A_i be a sequence of compact sets in M. Is it true that \bigcup \limits_{i=1}^{\infty} A_i = A is compact?

Solution

The answer is no since we can take A_i = \left[0,1-\frac{1}{i} \right] and M=[0, 1]. Hence

    \[A = \bigcup_{i=1}^{\infty} \left[0,1-\frac{1}{i} \right] = [0, 1)\]

which is clearly not compact.

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