Suppose that the sequence $\{a_n\}$ converges to $l$ and that sequence $\{b_n\}$ has the property that there is an index $N$ such that $\{a_n\}=\{b_n\}$ for all $n\geq N$. Show that $\{b_n\}$ converges to $l$.
So far I have used the definition of convergence on $\{a_n\}$. So $|a_n-l|<\epsilon$. Then I thought to use the Comparison Lemma where $C=1$, so we get $|b_n - l| \leq |a_n-l|<\epsilon$, which is true since $\{a_n\}=\{b_n\}$ for all $n\geq N$. That would clearly show what I needed to prove, but the problem is that I feel the last step is too much of a "leap of faith."
Any help is grateful, and thank you in advance.