thermodynamics - How to work out the relation between the "mean relative speed" and the "mean speed"?

I'm a freshman and am taking the general physics course. I just learned intro thermodynamics. One problem that really puzzles me is the calculation of "collision mean-free path", where calculating the mean relative velocity between gas molecules is needed. Our textbook simply gives a result $$\langle |v_r|\rangle =\sqrt2\langle |v|\rangle $$ without further explanation.
Here I am using angle brackets ($\langle \rangle $) to represent the "mean value" of what's inside. And note that all the velocities here are vectors, so I am using the absolute value symbols to get the "speed".

My professor has provided an explanation as follows:
Suppose that we select an arbitrary molecule A, with the velocity $v$ to the "stationary", as the reference frame. And suppose another arbitrarily selected molecule B has the velocity $v'$ to the "stationary". Therefore, in the reference frame A, B's velocity will be $(v'-v)$, which is just $v_r$, denoting B's "relative velocity" to A.
So we have: $$v_r=v'-v$$ Square both sides, $$|v_r|^2=|v'|^2+|v|^2-2v'\centerdot v$$ Now that we want to obtain the "mean value" of $v_r$ of an immense group of such "molecule B"s in a statistical sense,so my professor tried to work out the "mean value" of both sides: $$\langle |v_r|^2\rangle =\langle |v'|^2\rangle +\langle |v|^2\rangle -2\langle v'\centerdot v\rangle $$ It is plain to see (although there may be a lack of rigorousness) that, statistically $$\langle v'\centerdot v\rangle =0$$ and that $$\langle |v'|^2\rangle =\langle |v|^2\rangle $$ Therefore $$\langle |v_r|^2\rangle =2\langle |v|^2\rangle $$ Here comes the key part.From the above equation my professor concluded that $$\langle |v_r|\rangle =\sqrt2\langle |v|\rangle $$

However, I do not think this plausible step holds water. Because I think that for a statistical variable $x$, $\langle x\rangle ^2$ and $\langle x^2\rangle $ are not necessarily equal. (especially when I later learned something about Maxwell velocity distribution and found that for gas molecules the mean speed $|v|$ is actually smaller than the root mean square speed $\sqrt{\langle |v|^2\rangle }$.) So I think, instead of getting the result we want, the last step in fact gives $$\sqrt{\langle |v_r|^2\rangle }=\sqrt{2}\sqrt{\langle |v|^2\rangle }$$

This problem has been bothering me for several weeks and I want it fully explained, in an explicit and rigor way. I think only by using the knowledge of probability can a mathematically-convincing explanation be achieved. Unluckily I haven't learned much about probability and knows very little about relevant theories. Would anybody help me about this? Merci.