ISL 2015 A1

Suppose that a sequence $a_{1}, a_{2}, \ldots$ of positive real numbers satisfies $$a_{k+1} \geqslant \frac{k a_{k}}{a_{k}^{2}+(k-1)}$$ for every positive integer $k$. Prove that $a_{1}+a_{2}+\cdots+a_{n} \geqslant n$ for every $n \geqslant 2$.