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.