MHB Limits of Functions .... L&S Example 10.7 (2) ....

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I am reading "Real Analysis: Foundations and Functions of One Variable"by Miklos Laczkovich and Vera Sos ...

I need help with an aspect of Example 10.7 (2) ... Example 10.7 (2) reads as follows:View attachment 7252
In the above text, we read the following: "... ... Since whenever $$\lvert x - 2 \lvert \lt \frac{1}{2} , \lvert x - 1 \lvert \gt \frac{1}{2}$$ ... ... "Can someone please explain why:$$\lvert x - 2 \lvert \lt \frac{1}{2} \Longrightarrow \lvert x - 1 \lvert \gt \frac{1}{2}$$ ...Help will be appreciated ... Peter
 
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Peter said:
Can someone please explain why:$$\lvert x - 2 \lvert \lt \frac{1}{2} \Longrightarrow \lvert x - 1 \lvert \gt \frac{1}{2}$$ ...
More trickery with the triangle inequality! $$1 = |2-1| = |(2-x) + (x-1)| \leqslant |2-x| + |x-1| < \tfrac12 + |x-1|$$ and therefore $|x-1| > 1 - \frac12 = \frac12.$

Or to put it in everyday language, "the closer is to 2, the further is from 1".
 
Opalg said:
More trickery with the triangle inequality! $$1 = |2-1| = |(2-x) + (x-1)| \leqslant |2-x| + |x-1| < \tfrac12 + |x-1|$$ and therefore $|x-1| > 1 - \frac12 = \frac12.$

Or to put it in everyday language, "the closer is to 2, the further is from 1".
OMG ... I suppose I'll get the knack of these inequalities after a bit of practice :( ...

Thanks for your help ... you certainly took the mystery out of why the inequality held true ...

Peter
 
Equivalently, $$|x- 2|< 1/2$$ means that [math]-1/2< x- 2< 1/2[/math]. Adding 1 to each part, 1/2< x- 1< 3/2. Since x- 1 is always positive, we have |x- 1|= x- 1> 1/2.
 
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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