Note that $0< a_m \leq 1$ for all $m$. Suppose there's no such $N$, then the infinite series $\sum_{m=1}^\infty a_m$ converges. Then $a_m \to 0$ as $m\to\infty$. \[a_1 + \cdots + a_m \geq a_1^2 + \cdots + a_m^2 = \frac{1}{a_{m+1}} \to \infty\]gives contradiction!

Note that $\frac{1}{m} \leqslant a_m \leqslant 1$, which finishes by comparison with the harmonic series.