Hello I am just having a quick question

in the textbook intro to real analysis, during one of the limit examples the author notes,

if $$|x-c| \lt 1$$ then $$|x| \lt |c| +1$$

What rules are used to say this? how can I be rigour sure this is true.

For context, the c is just a constant. Thanks everyone

  • $\begingroup$ Because $$|x-c| \ge |x|-|c|$$ $\endgroup$ – peterwhy Jul 10 '15 at 23:38

$|x| = |x-c+c|$

By triangle inequality, $|x| \le |x-c|+|c| < |c| + 1$


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