Let $x$ and $y$ be two vectors in a Hilbert space $H$.Prove that $\left\|x+cy\right\|\geq\left\|x\right\|$ for all complex number $c$ if and only if $x$ and $y$ are orthogonal.
It's easy to show that if $x$ and $y$ are orthogonal,then the inequality is valid.For the converse conclusion, I think we can choose some special value of $c$ to obtain that $<x,y>=0$. But I don't know how to choose some proper $c$ to get result.
To elaborate on the suggestion in the comments:
Assume $A=\langle x, y \rangle \ne 0$. After replacing $y$ with $\alpha\, y$, with non-zero $\alpha \in \mathbb C$, you can assume that $A$ is (non-zero) real. At that point, square the inequality, and expand the lhs - you may take $c$ to be real to find your desired contradiction.
Hope this helps.