# Convergence in mean square - missing step

My question regards validating the legitimacy of representing an AR(1) as a MA($\infty$) process. In my notes this is done by showing convergence in mean square, saying that:

$\mathbb{E}[(\sum_{i=0}^{n-1}\phi^i\epsilon_{t-i}-Y_t)^2]=\mathbb{E}[\phi^{2n}Y_{t-n}^2]$

And then noting that:

$\phi^{2n}\gamma_0 \rightarrow 0$ as $n \rightarrow \infty$

The second step is fine. Could anyone show how the first equality is made? I get the feeling this shouldn't be too hard, but I just don't see it.

[For context, see 3.5.1 of these notes, which are similar]

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