I want to prove the fact that EM waves travel at a speed of light.

I would start with this equation:-

$$c= \lambda\nu$$

I am told that this fact proves that the all EM waves travel at the speed of light. I don't see why multiplying the wavelength times frequency give you the speed. Let me take an example and find the speed of $\gamma$ rays.

I would not like to use the formula above because it looks counterintuitive to me.

Instead I would use:-

$$\vec{v}=\frac{\vec{s}}{t}$$

Here I would take a few assumptions.

$$\vec{s}=\lambda$$

But I am not able to figure what should be my $t$. But by using the above formula and fact that the speed will be equal to speed of light I can say that:-

$$t=\lambda c$$

But the problem is that I do not want to use the above formula or fact.

So can anyone prove that:-

$c=\lambda\nu$

And how can we find the speed of light using the formula for average speed as I wrote above without using the $c=\lambda\nu$ and EM waves travel at speed of light.

Just one more thing how can we show that EM waves travel at the speed of light by using the fact $c=\lambda\nu$.