Prove that limit inferior is same as limit superior for a convergent sequence I was reading the book "Understanding Analysis" by Stephen Abbott on my own. I came across the following problem.
Let $(a_n)$ be a convergent sequence. 
Let $y_n$=sup{$a_k:k\geq n$}. Then lim sup $a_n$ = lim $y_n$.
Similarly define lim inf $a_n$.

Prove that lim inf $a_n$ = lim sup $a_n$ if and only if $(a_n$) converges and in that case all three share the same value.
I don't know how to prove it. I know that there are alternate definitions of lim sup and lim inf . So if anyone uses those definitions in their proof, please mention those clearly and if possible, also state how that definition is equivalent to the above definition . A proof which doesn't use metric spaces etc is preferred since I have not studied it yet.

Notice that $\limsup_n a_n = \ell$ is equivalent to say that:


*

*for all $\varepsilon>0$ eventually one has $a_k < \ell + \varepsilon$

*for all $\varepsilon>0$ frequently one has $a_k > \ell -\varepsilon$


where eventually means that the following property is true for all sufficiently large $k$ while frequently means that the following property is true for arbitrary large values of $k$ (I'm not sure the english translation is the correct one).
Formally: we say that $P(k)$ holds eventually if there exists $n$ such that $P(k)$ holds for all $k\ge n$. We say that $P(k)$ holds frequently if for all $n$ there exists $k\ge n$ such that $P(k)$ holds.
The definition of $\lim_n a_n = \ell$ is the same as above with frequently in the second line replaced by eventually while the definition of $\liminf_n a_n=\ell$ has frequently and eventually exchanged.
Notice that eventually implies frequently and you are done...
