The Riesz' Lemma is as follows:
Let $Y$ and $Z$ be subspaces of a normed space $X$ of any dimension (finite or infinite) such that $Y$ is closed (in $X$) and is also a proper subset of $Z$. Then for every real number $\theta$ in the open interval $(0,1)$, there is a point $z$ in $Z$ such that $$||z|| = 1$$ and $$||z-y|| \geq \theta$$ for every $y$ in $Y$.
Now we want to prove the following assertion:
If $Y$ is finite-dimensional, then we can even take $\theta$ to be equal to $1$ in the statement of the Riesz' Lemma.
How to prove this?