Bias, variance, and irreducible noise in prediction error

Suppose $y = f(x) + \varepsilon$ where $\varepsilon$ has mean $0$ and variance $\sigma^2$, independent of everything else. You fit a model $\hat{f}$ on random training data and predict at a fixed point $x$.

Show that the expected squared prediction error at $x$ splits into irreducible noise plus bias squared plus variance, and use it to explain overfitting.