There are variables $a_{1}, a_{2}, ..., a_{m}$. Each variable can be only natural number. Let $m \ge k$
Also each element satisfy : $a_{i}\le m$
Let $S:=a_{1}+a_{2}+...+a_{m}$
How big should be $S$ to be sure that there are $k$ elements such each of them is equal or greater than $k$
Regards
Imagine a situation where your desired conclusion does not hold, i.e., where at most $k-1$ of the $a$'s are $\geq k$. How big could $S$ be in such a situation?
$k-1$ of the $a$'s can be as big as $m$, while the remaining $m-k+1$ of the $a$'s can only be as big as $k-1$. When they're all as big as possible, what's $S$?
Now if $S$ is to be any bigger than that, you can't have the situation I've been considering, so your desired conclusion will hold.