Proof $GM$ (in dB) of $2nd$ order system is $\infty$
For a second order transfer function

$$G(s)=\frac{k}{s^2+2\zeta \omega s +\omega^2}$$

What are the real part of the poles?

$$\Re\{s_{p1},s_{p2}\}=-\zeta\omega$$

The closed loop transfer function from a constant feedback $h$ is

$$G(s)=\frac{k}{s^2+2\zeta \omega s +\omega^2+kh}$$

$$s_{p1},s_{p2}=\frac{-2\zeta\omega\pm\sqrt{-4\omega^2(1-\zeta^2)-4kh}}{2}$$

$$s_{p1},s_{p2}=-\zeta\omega\pm\sqrt{-\omega^2(1-\zeta^2)-kh}=-\zeta\omega\pm\sqrt{\Delta}$$

What are the real part of the poles?

For a system with poles which have imaginary parts
$$\Re\{s_{p1},s_{p2}\}=-\zeta\omega$$

It means that it doesn't matter how big $k$ is.

But if the poles are pure real numbers, as $h&gt;0$, by increase of the k, $\Delta$ decreases and only the upper bound of the poles reduce. So, the poles do not go to the right hand side of s-plane.