mass - Mathematical proof on helicity of a massive fermion is not Lorentz invariant

What is the mathematical proof that the helicity of a massive spin-$1/2$ fermion is not Lorentz invariant?

Something is Lorentz invariant (e.g., $P_\mu P^\mu$) if it commutes with all the generators of the Lorentz group. Assuming this statement to be correct, the helicity operator, therefore, must not commute with all the generators $\frac{1}{2}\sigma_{\mu\nu}$ of Lorentz transformation. I should also be able to verify that in the limit the mass $m\to 0$, a Lorentz transformation doesn't change the helicity.

Is this line of reasoning sufficient to prove the failure of Lorentz invariance of the helicity operator?