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I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...I am focuses on Chapter 3: Elements of Point Set Topology ... I need help regarding a remark of Apostol's made after Definition 3.2 and Theorem 3.3 ...Definition 3.2 and Theorem 3.3 read as follows:

View attachment 8477

View attachment 8478

In a note at the end of the proof of parts of Theorem 3.3 we read the following:

"... ... We also have

\(\displaystyle \mid \ \parallel x \parallel - \parallel y \parallel \ \mid \ \leq \ \parallel x - y \parallel\) ... Could someone please show me how top formally and rigorously prove that ...\(\displaystyle \mid \ \parallel x \parallel - \parallel y \parallel \ \mid \ \leq \ \parallel x - y \parallel\) ...

Help will be appreciated ...

Peter

View attachment 8477

View attachment 8478

In a note at the end of the proof of parts of Theorem 3.3 we read the following:

"... ... We also have

\(\displaystyle \mid \ \parallel x \parallel - \parallel y \parallel \ \mid \ \leq \ \parallel x - y \parallel\) ... Could someone please show me how top formally and rigorously prove that ...\(\displaystyle \mid \ \parallel x \parallel - \parallel y \parallel \ \mid \ \leq \ \parallel x - y \parallel\) ...

Help will be appreciated ...

Peter