$| x_n - x$ | = | $x_n$ | - | $x$ | is this right or is it less than or equal to.
the equality is in fact $| x_n - x$ | less than or equal to 3 how does this mean that $| x_n |$ smaller or equal to 3 + $|x|$, couldn't it not necessarily greater than 3 thanks
$$|x_n-x| \leq 3 \\ \implies -3 \leq x_n-x \leq 3 \\ \implies x-3 \leq x_n \leq x+3 $$ If you know that $x_n \geq 0$ then $x_n = |x_n|$ which means $$x-3 \leq |x_n| \leq x+3$$ It should also be clear that $x+3 \leq |x|+3$ so the inequality above becomes $$x-3 \leq |x_n| \leq |x|+3$$ If $x_n<0$ then you know $-x_n = |x_n|$. You can flip the inequality to capture this. $$x-3 \leq x_n \leq x+3 \\ \implies -(x+3) \leq -x_n \leq -(x-3) \\ \implies -x-3 \leq |x_n| \leq 3-x$$ Again, it should be clear that $3-x \leq 3+|x|$, so we can rewrite the inequality above as $$-x-3 \leq |x_n| \leq 3+|x|$$ Now we know for either case that $|x_n| \leq |x|+3$