If $ \lim_{n\to \infty} (a_n - b_n) = 0$, with $\{a_n\}$ converging, then $\{b_n\}$ converges to the same limit.

Here is a question I am working on:

Let $\{a_n\}$ be a sequence with limit $L$. Suppose that $\{b_n\}$ is a sequence such that for some positive integer $N$ we have $b_n$ = $a_n$ for every $n\ge N$. Prove that $\lim_{n\to \infty} b_n = L$.

Here is my solution, I think it seems too simple, but I would appreciate some feedback or some tips on a better direction:

We have that $b_n$ = $a_n$ for every $n\ge N$. Simply take the limit of both sides to yield:

$\lim_{n\to \infty}b_n = \lim_{n\to \infty}a_n = L.$

Hence $\lim_{n\to \infty} b_n = L$, as required.