Let $a_n$ be a decreasing sequence of non negative real numbers, show with the help of the Cauchy criterion for infinite series that $a_n\to 0$ as $n\to\infty$ if $\displaystyle\sum_{k=1}^{\infty} a_k$ converges.
Now, I think I have proved this already by using the standard argument of if a series converges to L then the sequence of partial sums convergence to L,
$\lim\limits_{n\to\infty} \displaystyle\sum_{k=1}^{n} a_k=L$
it then follows that
$\lim\limits_{n\to\infty} \displaystyle\sum_{k=1}^{n} a_k$=$ \lim\limits_{n\to\infty} ( a_n+ \displaystyle\sum_{k=1}^{n-1} a_k)$ = $\lim\limits_{n\to\infty} a_n$ + $\lim\limits_{n\to\infty} \displaystyle\sum_{k=1}^{n-1} a_k$
which leads to $L = \lim\limits_{n\to\infty} a_n$ + L which implies $\lim\limits_{n\to\infty} a_n $= 0
But I don't understand how I'd do this using the Cauchy criterion and for the specific scenario specified in the question?