MHB Proof of Apostol's Definition 3.2 and Theorem 3.3: Help Appreciated

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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

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

Help will be appreciated ...

Peter
 

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Hi Peter,

Peter said:
Could someone please show me how top formally and rigorously prove that ...
$$\mid \ \parallel x \parallel - \parallel y \parallel \ \mid \ \leq \ \parallel x - y \parallel$$ ...

Here are a few hints:
  1. Write $\|x\| = \|x-y +y\|.$
  2. Use the triangle inequality.
  3. Obtain a lower bound for $\|x-y\|$.
  4. Repeat the above with $\|y\|,$ obtaining a lower bound for $\|y-x\| = \|x-y\|.$
  5. Note that the two lower bounds differ by a negative sign only.
  6. Conclude that ${\large |}\|x\| - \|y\|{\large|}\leq \|x-y\|$ since $${\large |}\|x\|-\|y\|{\large |}=\begin{cases}\|x\|-\|y\| & \|x\|\geq \|y\|\\ \|y\| - \|x\| & \|y\|\geq \|x\|. \end{cases}$$
Let me know if anything is still unclear.
 
You are right to ask this question. It is called the "reverse triangle inequality". You can try to prove it yourself by using the ordinary triangle inequality to estimate both $\|x\| = \|(x - y) + y\|$ and $\|y\| = \|(y - x) + x\|$.

EDIT: Sorry, I did not see the reply by GJA and the system did not warn me when I submitted mine.
 
GJA said:
Hi Peter,
Here are a few hints:
  1. Write $\|x\| = \|x-y +y\|.$
  2. Use the triangle inequality.
  3. Obtain a lower bound for $\|x-y\|$.
  4. Repeat the above with $\|y\|,$ obtaining a lower bound for $\|y-x\| = \|x-y\|.$
  5. Note that the two lower bounds differ by a negative sign only.
  6. Conclude that ${\large |}\|x\| - \|y\|{\large|}\leq \|x-y\|$ since $${\large |}\|x\|-\|y\|{\large |}=\begin{cases}\|x\|-\|y\| & \|x\|\geq \|y\|\\ \|y\| - \|x\| & \|y\|\geq \|x\|. \end{cases}$$
Let me know if anything is still unclear.
Thanks GJA ...

Working through your post ...

Appreciate your help...

Peter
 
Janssens said:
You are right to ask this question. It is called the "reverse triangle inequality". You can try to prove it yourself by using the ordinary triangle inequality to estimate both $\|x\| = \|(x - y) + y\|$ and $\|y\| = \|(y - x) + x\|$.

EDIT: Sorry, I did not see the reply by GJA and the system did not warn me when I submitted mine.
Thanks for the guidance and help, Janssens

Peter
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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