why is $\left|xy\ log(\left|x\right|+\left|y\right|)\right|\leq\left|(\left|x\right|+\left|y\right|)log(\left|x\right|+\left|y\right|)\right|$?

Question

I should note that this was used by my book in order to show that the limit of $xy\ log(\left|x\right|+\left|y\right|)$ at $(0,0)$ is $0$.

After several attempts in vain, I plotted the function $\left | x \right |+\left | y \right |-\left | xy \right |$ and it's not even nonnegative everywhere!

Help please. Thanks in advance.

Answer 1

You can cancel the logarithm, so, as your comment already suggests, what you really want is $|xy|\le|x|+|y|$. As you noted, this is not true in general, but it's true if $x$ or $y$ is sufficiently close to $0$. If $|x|\le1$, then $|xy|=|x||y|\le|y|\le|x|+|y|$ (and analogously if $|y|\le1$).