How can I define the set of all subsequential limits of the following sequence: $x_n = ln(n) + cos(n)$? I have proved the following property: since $x_n$ is unbounded it has a divergent subsequence, whose limit is infinity, but what can I say about the set of all subsequential (as well as superior and inferior) limits?
Set of all subsequential limits of an unbounded sequence
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real-analysis
sequences-and-series
limits
1 Answers
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$\mathbf{Hint}$: if $\lim_{n\to\infty} x_n=\pm\infty$ or $\lim_{n\to\infty} x_n=L\in\Bbb R$, then $$\limsup_{n\to\infty} x_n=\lim_{n\to\infty} x_n=\liminf_{n\to\infty} x_n$$