Doubt in proof of of lemma 4.1.6

[jman...@gmail.com]

Hi,

I have a question about what is happening in this video at 00:14:00 -

http://www.youtube.com/watch?v=SRf5pxmq8Oo#t=840s.

I have a doubt in the proof of the reverse implication ( E[|X|] finite implies E[X] converges absolutely or finite). I think that it is true only when E[|X|] = 0. Because in real analysis, I have studied that if a sequence |xn| converges to a real number not equal to zero, then xn converges to a real number may not always be true. Ex - xn = (-1)^n. The sequence xn takes value -1 and 1 thus does not converge to any real number but sequence |xn|=1 which coverges to 1. I don't understand how to prove the reverse implication?.

Thanks!

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