# Prove the following statement involving limits of sequences

I am struggling to prove the following statement: suppose $lim_{n\rightarrow\infty}a_n=a$ and $lim_{n\rightarrow\infty}b_n=b$. If $a_n\lt b_n$ for $n\in\mathbb{N}$, then $a\lt b$.

So far, I have written the definitions of $lim_{n\rightarrow\infty}a_n=a$ and $lim_{n\rightarrow\infty}b_n=b$. I think that the proof is fairly straightforward, but am not yet sure how to construct it. Could you please suggest a hint?

• The statement is not true as you've written it. You can only conclude that $a\leq b$. – Zev Chonoles Apr 27 '15 at 15:20
• That's a helpful comment. – Caleb Owusu-Yianoma Apr 27 '15 at 15:28

## 1 Answer

This statement is not true generally. For a counter-example you can take $a_n=-\frac1n$ and $b_n=0$