Where does 1.3606 approximation ratio come from for vertex cover approximation? I was watching a coursera video on Approximation algorithms and I understood the 2-approximation algorithm.

Later, the professor asks if we can do any better. The lecturer went on to say that computing 1.3606 approximation is NP-hard in the following words at 8:14.
Here's the transcript for that part.

Actually, what is also interesting to note is that okay, maybe we still have some hope that we can get a little bit below two for the approximation ratio, but as the second staple that you see here shows is that it's not possible to get really close to one, but it's impossible to get something better than roughly 1.3, because already computing a 1.36 approximation as you see here, already that problem is NP-hard.


How does he skip from nobody found a 1.99-approximation algorithm
to getting 1.3606 is NP-Hard?
Where does 1.3606 come from?
The fact that nobody found a $1.99$ approximation doesn't necessarily mean that the problem of finding one is $\mathsf{NP}$-hard (e.g., it might be the case that that $\mathsf{P} \neq \mathsf{NP}$ and a polynomial-time $1.99$-approximation algorithm exists, we are just unaware of it).
Additionally, we can prove that finding a $1.3606$-approximation is a $\mathsf{NP}$-hard problem, i.e., if $\mathsf{P} \neq \mathsf{NP}$ no $1.3606$-approximation algorithm exists. This is not a direct consequence of the previous fact, but a separate result by Irit Dinur and Samuel Safra. The details can be found here.