When does a limit of a sequence equal both its lim sup and lim inf? Suppose $(a_n)_{n\ge1}$ is a positive (so non-zero), real sequence with $\lim_{n\rightarrow\infty}a_n=L$, where $L\in[0,\infty]$. Is this equivalent with $\limsup_{n\rightarrow\infty}a_n=\liminf_{n\rightarrow\infty}a_n=L$? In particular, does this hold for the infinity case? If so, why? Thank you!

The $\displaystyle\liminf_{n \to +\infty} a_n$ denotes the smallest adherence value of the sequence $(a_n)_n$. The $\displaystyle\limsup_{n \to +\infty} a_n$ denotes the greatest adherence value of the sequence $(a_n)_n$. Recall that $\ell$ is an adherence value of $(a_n)_n$ if there is a subsequence of $(a_n)_n$ which converges to $\ell$. Thus, if the limit inf is equal to the limit sup then the whole series converges to this common limit. The reverse is also true.
