Partial sums of divergent series

Question

Let $\sum a_n$ be a divergent serie of positive terms. Prove that for each positive integer $m$ there is $n>m$ such that

$$a_{m+1}+\cdots a_n> a_1+\cdots + a_{m}.$$

I tried to use that the partial sum sequence is not Cauchy but unsuccessful.

Answer 1

Let $N = a_1 + \cdots + a_m$. Is there ever a partial sum that exceeds $2N$?

Answer 2

Assume the contrary. Then the sequence $s_k=\sum_{j=1}^k a_{n+j}$ is bounded amd increasing.