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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?

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    $\begingroup$ The statement is not true as you've written it. You can only conclude that $a\leq b$. $\endgroup$ – Zev Chonoles Apr 27 '15 at 15:20
  • $\begingroup$ That's a helpful comment. $\endgroup$ – Caleb Owusu-Yianoma Apr 27 '15 at 15:28
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This statement is not true generally. For a counter-example you can take $a_n=-\frac1n$ and $b_n=0$

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