If $\lim_{n\to \infty} a_n = A$, $\lim_{n\to \infty} b_n = B$ and $A

If $\lim_{n\to \infty} a_n = A$, $\lim_{n\to \infty} b_n = B$ and $A

If $\lim _{n\to \infty} a_n = A$, $\lim _{n\to \infty} b_n = B$ and $A<B$,
prove $\lim _{n\to \infty}\max\{a_n,b_n\} = B$
Side question: Does the $A<B$ imply $a_n < b_n$?
Since we know $a_n$ converges to $A$, doesn't the problem kind of suggest
$\max{a_n,b_n} = b_n$ (as we also know $A\not = B$, so if the max were
$a_n$ the limit would have to be A, but then the problem is futile)?
[Not homework, just practice for learnings sake :-)]