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Let $(a_n)_{n\in \mathbb{N}}$ and $(b_n)_{n\in \mathbb{N}}$ be two real sequences that satisfy $a_n\geq b_n, \forall n \in \mathbb{N}$ and converge to some $a,b$, respectively.

Is it always true that $a \geq b$?

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marked as duplicate by Martin Sleziak, R_D, JonMark Perry, M. Vinay, Davide Giraudo real-analysis Jul 7 '16 at 9:43

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  • $\begingroup$ Yes. What can you say about the limit of $(a_n-b_n)$? $\endgroup$ – David Mitra Jun 29 '13 at 12:55
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    $\begingroup$ Suppose you have $a_n \geqslant b_n$, $a_n \to a$, $b_n \to b$ and $a < b$. Try to find a contradiction. $\endgroup$ – Daniel Fischer Jun 29 '13 at 12:55
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Yes. If you assume that $a < b$, you can take $\varepsilon = (b-a)/2$ and use the definition of the limit of a sequence to come up with a contradiction, i.e. find an $n$ such that $a_n < b_n$.

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