<s>[INST] Proving a certain inequality I'm trying to prove the following inequality by purely algebraic means
$$\frac{|x-y|}{1+|x-y|}\leq \frac{|x-z|}{1+|x-z|}+\frac{|z-y|}{1+|z-y|}.$$
My arguement is as follows: 
Notice that, $|x-z|+|z-y| \geq |x-y|$ and from this inequality we can write $1+|x-z|+|z-y| \geq 1+|x-z| $ and $1+|x-z|+|z-y| \geq 1+|z-y|$. Now, 
$$\frac{|x-y|}{1+|x-y|} \leq \frac{|x-z|+|z-y|}{1+|x-z|+|z-y|}\leq \frac{|x-z|}{1+|x-z|}+\frac{|z-y|}{1+|z-y|}.$$
Is this correct? 
 [/INST] We have
\begin{align}
\frac{|x - y|}{1 + |x - y|} &= 1 - \frac{1}{1 + |x - y|}\\
& \le 1 - \frac{1}{1 + |x - z| + |z - y|}\\
& = \frac{|x - z| + |z - y|}{1 + |x - z| + |z - y|}\\
& = \frac{|x - z|}{1 + |x - z| + |z - y|} + \frac{|z - y|}{1 + |x - z| + |z - y|}\\
& \le \frac{|x - z|}{1 + |x - z|} + \frac{|z - y|}{1 + |z - y|}.
\end{align}
</s>