Let $(X,\|.\|)$ be a Banach space over the real line. Let $x\neq 0$ and $y\neq 0$ be in $X$, then $x$ is said to be orthogonal to $y$ if $\|x+\lambda y\|\geq\|x\|$ for every real number $\lambda$. It is said that given any $x$ and $y$ there exists a real number $\alpha$ such that $x$ is orthogonal to $\alpha x + y$. Can anyone tell how?

For reference see Corollary 2.2 in this article. I am not able to prove the existence in the corollary, but am able to prove the rest of it.

I can see it diagrammatically, that is we need to show that $x$ is orthogonal to a point in the line passing through $y$ and in the direction of $x$. But I am not able to find this $\alpha$ rigourously