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Retraction proof

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Let A_0 be contained in A_1 contained in A_2 and so on be a nested sequence of subspaces of X such that the union of all A_n is X and such that An contained in the interior of A_(n+1). Suppose for each n, there is a retraction r_n:A_(n+1) to An. Prove there is a retraction r: X to A_0.

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The solution provides an example of proving the existence of a retraction.

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