Subsequence of a sequence 
If $\lim_{n\rightarrow\infty}{\langle a_n \rangle} = a$ and $\langle a_{in} \rangle$ is any subsequence of $\langle a_n \rangle$, then $\lim_{n\rightarrow\infty}{\langle a_{in} \rangle} = a$, but the opposite is not (necessarily) true.

I am trying to understand the above theorem. However, I am struggling to come up with any examples that prove this theorem and example the prove that the opposite to this theorem is not true. Are there any clear examples that show this theorem.

That means , if a sequence converges to a limit then all sub-sequence of it converges to the same limit. But if two or more sub-sequences are convergent then the sequence need not be convergent.
For example , consider the sequence $\{x_n\}=\{(-1)^n\}$. It has two sub-sequences : $\{1,1,\cdots\}$ which converges to $1$ and $\{-1,-1,\cdots\}$ which converges to $-1$ but the sequence $\{x_n\}$ is NOT convergent.
