In trying to solve the following inequality in R:

|x+y| |y+z|+|z+x| $\displaystyle \leq$ |x+y+z| |x| +|y| + |z|,i tried a proof by contradiction.

So suppose the above inequality is not true ,then we have:

|x+y| |y+z|+|z+x| > |x+y+z| |x| +|y| + |z|

....................or 2|x| +2|y| +2|z| > |x+y+z| +|x| +|y| +|z|

.....................or |x| +|y| + |z|> |x+y+z| and here is where i get stuck because i cannot derive a contradiction