Question: Show that if and only if there is a sequence 0 < e n 0 such that For the part e n 0, show that,
if and only if there is a sequence 0 < en 
0 such that
For the part en 
0, show that, for every e > 0, there exists N = N(e) such that k ³ and n ³ imply 
(|Xk €“ X| ³ e) à 
(|Xk €“ X| ³ ek) 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 nm †‘ ¥ as m †’ ¥ such that
Finally, for nm £ k < nm +1, set ek = 1/m and show that
a.s. X "X
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