If the sequence $\left\{a_n\right\}$ has no two subsequence converging to two different limits then the sequence has a limit?
Can I understand the question in the way that: If all the convergent subsequences of $\left\{a_n\right\}$ converge to the same limit, then the sequence has a limit? Does this question has a counter example?
As stated, the question has some trivial counter-examples, as angryavian mentioned, since $(a_n)$ may not have any convergent subsequences. In that case the condition "$\{a_n\}$ has no two subsequences that converge to two different limits" is satisfied trivially; so are conditions of the form "if all the convergent subsequences...". The sequence $a_n := n$ then gives a trivial counter-example.
Note that $\overline{\lim} a_n$ and $\underline{\lim} a_n$ are always two particular subsequences of $(a_n)_n$.
If $a_n$ is bounded, these are monotonic sequences in a compact thus always converge.
So if you require that no two subsequences converge to different values, then it is also true for these two, and $\underline{\lim}=\overline{\lim}$ means $a_n$ converge.
Now if $a_n$ is not bounded, as stated there may not be converging subsequences [e.g. $a_n=n$].
Yet if you consider compactified $\mathbb R\cup\{-\infty,+\infty\}$ then the extended condition still impose $\underline{\lim}=\overline{\lim}$ so $a_n$ diverges to $\pm\infty$. This is a result stronger than just unbounded.
