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.