Convergence of $\sin (n_k x)$

Question

If I have a set $E$ which is a set of points $x\in(-\pi,\pi)$ such that for an increasing sequence $\{ n_k \}$ a sequence of functions $\sin(n_k\,x)$ converges, where do I start proving that measure of $E$ is zero?

Answer 1

Hint: Suppose $\sin (n_kx) $ converges pointwise to some $f$ on a set $E\subset (-\pi,\pi)$ of positive measure. Observe that $\int_Ef^2 = \lim_{k\to \infty} \int_E f(x)\cdot (\sin n_kx) \, dx = \lim_{k\to \infty} \int_E (\sin n_kx)^2 \, dx$