Is the root-mean-square velocity of gas particles always faster than the average velocity? I've done a couple of calculations with both and that seems to be the case; however can you make a formal mathematical proof of it? (i.e. maybe using Cauchy-Schwarz)
The classical analysis is known as the Maxwell-Boltzmann distribution.
The RMS velocity is about 108% of the average velocity.
Root-mean-square velocity is a scalar, average velocity is a vector. I am not sure how one can compare them. Maybe you meant the absolute value of average velocity?
EDIT (09/25/2018): So the OP confirmed that in his comment. So you can consider the difference between the squares of the rms velocity and average velocity $(\sum_i V_i^2)/n-(\sum_i V_i)^2/n^2$, which is a quadratic form, and find its minimum requiring that all derivatives with respect to $V_i$ vanish.