Putnam 2000 B3

Let $f(t)=\sum_{j=1}^N a_j \sin(2\pi jt)$, where each $a_j$ is real and $a_N$ is not equal to 0. Let $N_k$ denote the number of zeroes (including multiplicities) of $\frac{d^k f}{dt^k}$. Prove that $$N_0\leq N_1\leq N_2\leq \cdots \text{ and } \lim_{k\to\infty} N_k = 2N.$$ [Editorial clarification: only zeroes in $[0, 1)$ should be counted.]