Self-study question
Given $(y_i, x_i)$, $i = 1, . . . , n$, where $y_i \in \mathbb{R}$ and $x_i ∈ R ⊂ \mathbb{R}^p$.
Show that $\displaystyle \sum_{i:x_i \in R_1}(y_i − \hat{y}_{R_{1}})^2 +\sum _{i:x_i \in R_2}(y_i − \hat{y}_{R_{2}})^2 \leq \sum_{i:x_{i} \in R}(y_i − \hat{y}_{R})^2$
where $\hat{y}_{R_{j}}$ is the mean response for the training observations within the $j$th region, and $R = R_1 \cup R_2, R_{1} \cap R_{2} = \{ \}$. Also,
$$\hat y_{R_1}=\frac{1}{n_1}\sum_{i:x_i\in R_1} y_i,$$ $$\hat y_{R_2}=\frac{1}{n_2}\sum_{i: x_i\in R_2} y_i,$$
where $n_{1}$ and $n_{2}$ are number of observations in $R_{1}$, and $R_{2}$, respectively.
My approach:
So, I solved it in a very long way, but I was hopping that someone will propose a much simpler solution (Note: I skipped several steps between the first and the second line)
\begin{align*} \sum_{i=1}^n (y_i - \bar{y})^2 &= \sum_{i=1}^n y_i^2 - n \, \bar{y}^2 \\ &= \sum_{i=1}^{n_{1}} (y_i -\bar{y}_{1})^{2}+\sum_{i=n_{1}+1}^{n} (y_{i} - \bar{y}_{2})^{2}+ \frac{n_{1} \, n_{2}}{n} \, (\bar{y}_{1}-\bar{y}_{2})^{2}\\ & \geq \sum_{i=1}^{n_{1}}(y_i -\bar{y}_{1})^{2}+\sum_{i=n_{1}+1}^{n} (y_{i} - \bar{y}_{2})^{2} \end{align*}
