Limits and Continuity - Absolute Value Technicality ....

[Math Amateur]

I am reading Manfred Stoll's Book: "Introduction to Real Analysis" ... and am currently focused on Chapteer 4: Limits and Continuity ...

I need some help with an inequality involving absolute values in Example 4.1.2 (a) ...

Example 4.1.2 (a) ... reads as follows:

In the above text we read ...

"... If $\mid x - p \lvert \ \lt 1$ then $\mid x \mid \lt \mid p \mid + 1$ ... "Can someone please show me how to rigorously prove the above statement ...

Peter

 

[Orodruin]

Just use the triangle inequality.

 

[member 587159]

$|x| = |x-p+p| \leq |x-p| + |p| \Rightarrow |x| - |p| \leq |x-p| < 1 \Rightarrow |x| < 1 + |p|$