This question from MWG 8.B.7

Any strictly dominant strategy must be a pure strategy.

How can I show this?

My explanation is as follows:

Suppose we have a strictly dominant strategy, $\sigma_i$ . Suppose further that $\sigma_i$ is not a degenerate pure strategy. Then $\sigma_i$ cannot strictly dominate any pure strategy for which $\sigma_i$ specifies playing with positive probability. Hence $\sigma_i$ cannot be strictly dominant. Thus, $\sigma_i$ must be a degenerate pure strategy if it is to be strictly dominant.

But I guess this is just an explanation what I thought.

How can I prove this sentences mathematically?