Suppose we are given a sequence of nonnegative real numbers:
$a_1\geq a_2\geq a_3\geq\dots\geq 0$
such that
$\lim_{n \to \infty} a_n=0.$
Assume that
$\lim_{k\to\infty} (a_{n_k}\ln n_k)=0$
for some sequence $(n_k)_{k=0}^\infty$ of natural numbers.
Is it true that
$\lim_{n\to\infty} (a_{n}\ln n)=0 $
also holds?
Keep in mind that the sequence $a_n$ need not to be strictly monotonic.