Prove a sequence S_n converges iff its tail S_n+M converges.
you just have to use the definition of convergence.
Assume $\displaystyle S_n \rightarrow a$. we will show that $\displaystyle S_{n+M} \rightarrow a$.
$\displaystyle S_n \rightarrow a$ gives us:
$\displaystyle \forall \epsilon >0, \exists N \text{ such that } n>N \Rightarrow |S_n-a|<\epsilon$
To prove $\displaystyle S_{n+M} \rightarrow a$, that is:
$\displaystyle \forall \epsilon>0, \exists N^{'} \text{ such that } n>N^{'} \Rightarrow |S_{n+M}-a|<\epsilon$. We easily see that $\displaystyle N^{'}=N-M$ does the job.
Now can you try and do the 'only if' part.