TITLE: Inequality between two sequences preserved in the limit?
QUESTION [5 upvotes]: 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 $a \geq b$?

REPLY [5 votes]: 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 $a_n < b_n$.