Let $f$ be a positive, continuous, decreasing function such that $a_n=f(n).$ If ${\displaystyle \underset{n=1}{\overset{}{}}}{a}_n$ converges to $k,$ which of the following must be true?

$($A$)\underset{n}{lim}{a}_n=k$

$($B$){\displaystyle \underset{1}{\overset{n}{}}}f(x)dx=k$

${\displaystyle \underset{1}{\overset{}{}}}f(x)dx$ diverges

$($D$){\displaystyle \underset{1}{\overset{}{}}}f(x)dx$ converges

$($E$){\displaystyle \underset{1}{\overset{}{}}}f(x)dx=k$