For a normed vector space, is $\|x-y\| \leq \|x\|+\|y\|$ true? I have a question about an inequality in normed vector spaces and I want to know if my proof is correct.
Claim: 
Let $X$ be a normed vector space. Then \begin{equation} \|x-y\| \leq \|x\|+\|y\|\end{equation} for all $x,y \in X$.
Proof: Using the triangle inequality and the fact that $\|z\|=\|-z\|$, we have \begin{equation}
\begin{split}
\|x-y\|&= \|x+(-y)\|\\
&\leq \|x\|+\|-y\| \\
&= \|x\|+\|y\|
\end{split}
\end{equation}
Thanks for your answers!

As confirmed in the comment by Axel, your proof is correct.
