How do you prove $\lvert x - y \rvert < 1$ then $\lvert x\rvert<\lvert y\rvert +1 $?
I know this proof has the form of the triangle inequality, but I can't seem to figure it out. This is from Kenneth Ross 17.1
2026-02-23 13:30:50.1771853450
How do you prove $\lvert x - y \rvert < 1$ then $\lvert x\rvert<\lvert y\rvert +1 $?
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$|x| = |x-y+y| \leq |x-y|+|y|\stackrel{|x-y|<1}{\implies} |x| < 1+|y|$
More generally, this trick shows $|x|-|y|\leq |x-y|$ and by symmetry, $|y|-|x|\leq |x-y|$ and therefore, $||x|-|y||\leq |x-y|$ which is a common way to prove that absolute value is a (uniformly) continuous function.