Riesz's Lemma, which is 2.5-4 in Erwin Kreyszig's Introductory Functional Analysis With Applications, is as follows:
Let $Y$ and $Z$ be subspaces of a normed space $X$ (of any dimension), and suppose that $Y$ is closed and is a proper subset of $Z$. Then for every real number $\delta \in (0,1)$, there is an element $z\in Z$ such that $\Vert z \Vert =1$ and $\Vert z-y \Vert \geq \delta$ for all $y \in Y$.
Now Problem 7 in the problem set immediately following Section 2.5 in Kreyszig is as follows:
If $\dim Y < \infty$ in Riesz's Lemma, show that one can even choose $\delta = 1$.
I've read and I think I've understood fully the proof of the Lemma with $\delta \in (0,1)$, and I'm unable to figure out to modify that particular proof to include the case when $\delta = 1$. Nor am I able to give an independent proof of the assertion made in Problem 7.
Can anybody please help?