A sequence $a_1,a_2,\ldots$ of real numbers has the following properties:

(i) $|a_1+a_2+\ldots +a_k|\leq 1$ for every $k$;

(ii) $|a_k-a_{k-1}|\leq 1/k$ for every $k\geq 2$.

Suppose that $\displaystyle |a_k|\geq \frac{c}{\sqrt{k}}$ for infinitely many $k$. Prove that $c\leq \sqrt{2}$.