Upper Bound of a Subseries of a Positive Convergent Sequence

Question

Let $(a_n) \to 0$ and that $a_n >0, \forall n$. I am trying to show that: $$ \exists (a_{n_k}): \sum_{k=1}^{\infty}a_{n_k}<1 $$ Thought: I can show that $(a_n)$ has a strictly decreasing subsequence. Because $(a_n) \to 0$, this subsequence can be constructed to have an upper bound as small as needed. I think this has to do with this question. Other than that, I am not sure how to tackle the problem.

Answer 1

Hint

Since $a_n\to 0$, there exists $n_1<n_2<\dotsb$ such that $$ 0<a_{n_k}< \frac{1}{4^k}. $$ Now sum over $k$ and use the formula for a geometric series.