TITLE: Proof of Riesz Lemma
QUESTION [6 upvotes]: Here is the proof of the Riesz Lemma. 
When the author show that $\| z-y \| \geq \theta$, I don't understand the following step:
$$\dfrac{1}{\| v-y_0 \|} \| v-y_0 - (\| v-y_0 \|)y \| \geq \dfrac{\alpha}{\| v-y_0 \|}$$
How to obtain the above step?

REPLY [4 votes]: $v - y_0 - (||v-y_0||)y$ is of the form $v - y'$ with $y' \in Y$ (as $Y$ is a subspace, and $y, y_0 \in Y$).
The norm of it is thus bounded below by $\alpha$ by the definition of $\alpha$ as $\inf_{y \in Y} ||v - y||$.