How to turn $|a_n| < 1 + |L|$ into $L -1 < a_n < L + 1$?

Question

So in class, we went over the proof of convergent sequences are bounded, I get pretty much everything except: $|a_n| < 1 + |L|$ part. How to turn $|a_n| < 1 + |L|$ into $L -1 < a_n < L + 1$? Where did the absolute value go?

Answer 1

Presumably $n$ is large enough that $|a_n - L| < 1$ is true, since $a_n$ converges to $L$. Then $$-1 < a_n - L < 1$$ so $$L - 1 < a_n < L + 1.$$ But in general, given two arbitrary quantities $x$ and $y$ it is not true that $$|x| < 1 + |y|$$ implies $$y-1 < x < y+1.$$ (To see why, just take $x = -y > 1$.)