For a particle to exhibit simple harmonic motion, the force acting on it always has to be $\vec{F} = -k\vec{x}$. But here:
Let's say it has some initial elongation. Now, the only way it would have $\vec{F} = -k\vec{x}$ is if it was pulled down by some distance $x$ since the tension in the string connected to the string can only pull, not push. But when the block is moving up and passes the mean position, there's no restoring force $-kx$ on it(A string can't exert such a force).
How does this system show SHM?