Saturday, 27 December 2014

limits - If limxtoinftyxn=liminfntoinftyxn=limsupntoinftyxn=infty does it exist a convergent subsequence of xn?

If lim does it exist a convergent subsequence of x_n? I've learned that the limit of a convergent subsequence is called accumulation point,\limsup as the greatest accumulation point and \liminf as the smallest accumulation point. Then if \limsup and \liminf are -\infty then all limits of all subsequences of x_n are between -\infty and -\infty i.e. any of the sequences converges. I'm not really sure if this argument is correct. Could you help me please?

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