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I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ...
I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ...
I need some help with an aspect of the proof of Theorem 11.1.11 ...
Garling's statement and proof of Theorem 11.1.11 reads as follows:
View attachment 8945
View attachment 8946Near the end of Garling's proof above we read the following:
" ... ... Suppose that $$f,g,h \in B_X(S)$$ and that $$s \in S$$. Then
$$d(f(s), h(s)) \le d(f(s), g(s)) + d(g(s), h(s)) \le d_\infty (f, g) + d_\infty (g, h)$$ ... ... ... (1)
Taking the supremum, $$d_\infty (f, h) \le d_\infty (f, g) + d_\infty (g, h)$$ ... ... ... "Now (1) is true for arbitrary s and so it is true for all $$s$$ including the point for which $$d(f(s), h(s))$$ is a maximum ... if a maximum exists ...But in the case where a maximum does not exist ... how do we know that taking the supremum preserves inequality (1) ...Hope someone can help ...
Peter
I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ...
I need some help with an aspect of the proof of Theorem 11.1.11 ...
Garling's statement and proof of Theorem 11.1.11 reads as follows:
View attachment 8945
View attachment 8946Near the end of Garling's proof above we read the following:
" ... ... Suppose that $$f,g,h \in B_X(S)$$ and that $$s \in S$$. Then
$$d(f(s), h(s)) \le d(f(s), g(s)) + d(g(s), h(s)) \le d_\infty (f, g) + d_\infty (g, h)$$ ... ... ... (1)
Taking the supremum, $$d_\infty (f, h) \le d_\infty (f, g) + d_\infty (g, h)$$ ... ... ... "Now (1) is true for arbitrary s and so it is true for all $$s$$ including the point for which $$d(f(s), h(s))$$ is a maximum ... if a maximum exists ...But in the case where a maximum does not exist ... how do we know that taking the supremum preserves inequality (1) ...Hope someone can help ...
Peter