# Mixed strategy in extensive form games with complete and perfect information

I saw the lemma:

"In extensive form games with complete and perfect information, any mixed strategy for player i will result in a lower or equal utility for player i compared to some pure strategy available to player i."

I understand it intuitively, that a mixed strategy decision, would be sub-par to the best-response I can take, but I don't know how to prove it.

Given a profile of the other players' strategies $$\sigma_{-i}$$, player $$i$$'s utility from playing a mixed strategy $$\sigma_i$$ is given by: $$$$u_i(\sigma_i,\sigma_{-i})=\sum_{s_i\in S_i}\sigma_i(s_i)u_i(s_i,\sigma_{-i})$$$$ where $$\sigma_i(s_i)$$ is the probability of playing pure strategy $$s_i$$ according to $$\sigma_i$$. Now let $$s_i^*\in\mathop{\arg\max}_{s_i\in S_i}\;u_i(s_i,\sigma_{-i}).$$ Such a maximum must exist because $$S_i$$ is finite. It follows that, for all $$\sigma_i$$, $$u_i(s_i^*,\sigma_{-i})\ge\sum_{s_i\in S_i}\sigma_i(s_i)u_i(s_i,\sigma_{-i}) \tag*{\blacksquare}$$