Prove of inequality under a Hilbert space.

Use paralleogram law for $\frac{x}{2}$ and $\frac{y}{2}$ to obtain $||\frac{x+y}{2}||^2 + ||\frac{x-y}{2}||^2 = \frac{1}{2}||x||^2 + \frac{1}{2}||y||^2$ and so you get $1$ in the right hand side. Since the LHS is a sum of two non-negative terms, you get the desired inequality since $x\neq y$ .

So this is my final answer for any interested party answer and thank you @joy

$\|\frac{x+y}{2}\|^2 \leq \|\frac{x+y}{2}\|^2 + \|\frac{x-y}{2}\|^2 = \frac{1}{2}\|x\|^2+\frac{1}{2}\|y\|^2$

and since $x \neq y$:

$\|\frac{x+y}{2}\|^2 < \frac{1}{2}\|x\|^2+\frac{1}{2}\|y\|^2=\frac{1}{2}r^2+\frac{1}{2}r^2=r^2$

Hence:

$\|\frac{x+y}{2}\| < r$