A sequence is a subsequence of if and only if there is an increasingfunction such that For example, is thesequence whose terms are just the even-numbered term of the sequence(a) Let Describe the subsequences and(b) Prove that if a sequence x converges to L then for every real thereexists a subsequence y of x such that for all(c) Prove that if converges to L and is a subsequence of x, then y convergesto L.(d) Prove that if x contains two convergent subsequences y and z,and and then x diverges.

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