If $\lim_{x \to \infty} x_n = \liminf_{n \to \infty} x_n = \limsup_{n \to \infty} x_n= -\infty$ 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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