Prove that a sequence which converges at L, still converges at L when a fixed positive integer is added to the variable.

Question

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.

Answer 1

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.

Answer 2

You are on the right track.

Please fix the typo '$n<N$ 'to $n>N$ Notice that if $n+p>N\iff n>N-p$

Answer 3

Properties definitely satisfied by $a(n)$ are definitely satisfied by $a(n+p)$. Moreover, viceversa holds.