A problem regarding neighborhood of a point in Normed linear space
Hint: Sketch of Proof
Define $z=\alpha x+\beta y$. Since $z$ is a linear combination of $x$ and $y$, then all the three vectors $x,y,z$ are coplanar and since they are all unit vectors, then they lie on the same unit circle. Hence $B_\delta$ intersects that unit circle exactly at its own center (i.e. center of $B_\delta$) and a non-zero part of the unit circle, falls inside $B_\delta$. Now find a point $\tilde \alpha x+\tilde \beta y$ on the intersection of the unit circle and $B_\delta$ such that $\tilde\beta<\beta$.