Prove that a sequence which converges at L, still converges at L when a fixed positive integer is added to the variable. Here is the problem I am attempting to solve/prove:
Let $(a_n)$ n∈N be a sequence that converges to L and let p be a fixed positive integer. Prove that the sequence $(a_{n+p})$ n∈N converges to L.
I've started by saying $\epsilon >0$ and $|a_{n+p} -a|<\epsilon$. 
And I think I need to find an N, $n>N$ which implies $|a_{n+p} -a|<\epsilon$ ?
If I'm going in the correct direction, I'm not really sure where to go from there so a tip would be great.
Also, I apologize if I am not using this website correctly. Any critique/tips would be greatly appreciated. 

Yes, you are going in the right direction.  You are told that $a_n \to L$, so if you give the person who asked the problem an $\epsilon \gt 0$ they have to be able to give you an $N$ such that for all $n \gt N, |a_n - L| \lt \epsilon$  If somebody asks you the question for $a_{n+p}$ you turn the same $\epsilon$ over to the person with $a_n$.  Now note that $n+p \gt n \gt N$ so your error is less than $\epsilon$, too and you are done.
