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So I have been learning about sequences recently and I am struggling with the following problem:

Given two convergent sequences $(a_n)_{n \geq 0}$ and $(b_n)_{n \geq 0}$, where $a_n < b_n$ for all $n$, prove that $\lim a_n \leq \lim b_n$ follows.

It seems obvious that this would be true, but I can't just quite prove it. I thought maybe looking at the difference of the two sequences $b_n - a_n$ would be a good idea. Since $b_n > a_n$ for all $n$ this sequence would contain only positive numbers. If I could somehow show, that this always converges to some $d \geq 0$ than I thought this would prove it, however I haven't really been able to make some progress there so now I'm back to square one. I would really appreciate some help.

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Let $d_n\geq 0$. If $d=\lim d_n<0$ then for $\varepsilon =-d/2>0$ there is $N$ such that for $n>N$ $|d_n-d|<-d/2$. But $|d_n-d|=d_n-d\geq -d> -d/2$. A contradiction.

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  • $\begingroup$ I think you made a small error at the beginning. Shouldn't it be $d = \lim d_n < 0$? $\endgroup$ – roblox99 Mar 7 at 18:13
  • $\begingroup$ Yes, I fixed. Is it clear now? $\endgroup$ – user37274 Mar 7 at 18:14
  • $\begingroup$ Thank you for your answer. Looking at your proof it all seems so obvious and easy now. $\endgroup$ – roblox99 Mar 7 at 18:15
  • $\begingroup$ That is very good! $\endgroup$ – user37274 Mar 7 at 18:16
  • $\begingroup$ However I only have one question left. Would this proof still be valid if if $d_n$ didn't have a limit? Or is it guaranteed to have one since its the difference of two convergent sequences? $\endgroup$ – roblox99 Mar 7 at 18:19

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