Show that if and only if there is a sequence 0 < e_n  0 such that

For the part e_n  0, show that, for every e > 0, there exists N = N(e) such that k ³ and n ³ imply  (|X_k €“ X| ³ e) Í  (|X_k €“ X| ³ e_k)  and then use Theorem 4 suitably. For the part , use Theorem 4 in order to conclude that

Applying this conclusion for m ³ 1, show that there exists a sequence n_m †‘ ¥ as m †’ ¥ such that

Finally, for n_m £ k < n_{m +1}, set e_k = 1/m and show that

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