Let $t_{n-1, \alpha/2}$ denote the critical point (upper tail probability of $\alpha/2$) of the Student's T distribution with $n-1$ degrees-of-freedom. I am examining the function:
$$w(\alpha, n) = \frac{t_{n-1, \alpha/2}}{\sqrt{n}} \quad \quad \quad \text{for all } 0 <\alpha <1 \text{ and } n > 1.$$
From statistical reasoning, plus graphical analysis of the function, I am convinced that this function is monotonically decreasing in $n$ (which is equivalent to $\sqrt{n}$ increasing faster than $t_{n-1, \alpha/2}$). I would like to prove this fact by showing that:
$$\frac{\partial w}{\partial n} (\alpha, n) < 0.$$
I have tried various methods to do this (implicit differentiation, looking at finite differences, etc.) and I have been unable to crack it so far. Would any of you mathematics gurus like to have a go?