It may be very stupid question, please bear with me.
Recall we define $|a|=a$ if $a\geq 0$ and $|a|=-a $ if $a\leq 0$
Now we have, $$-a\leq |a|\leq a$$ and $$-b\leq|b|\leq b$$
so using these two I get $$-(a+b)\leq |a|+b|\leq a+b$$
That imply $$||a|+|b||\leq |a+b|$$
But this is incorrect as take $a=1,b=-1$, we get $LHS=2$, but $RHS=0$
What have I done wrong?
$$ -a\leq |a|\leq a $$ does not hold in general, it is wrong for negative numbers. What you probably meant is $$ -|a|\leq a\leq |a| \\ -|b|\leq b\leq |b| $$ and adding those inequalities gives $$ -|a| -|b|\leq a+b\leq |a| + |b| $$ which is the well-known triangle inequality $$ |a+b| \le |a| + |b| \, . $$