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I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...
I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ...
I need some help in understanding the proof of Proposition 3.7 ...Proposition 3.7 and its proof read as follows:View attachment 9509
In the above proof by Andrew Browder we read the following:
" ... ... Clearly $$A\leq f(t) \leq B$$ since $$f$$ is increasing ... ... "
Can someone demonstrate, formally and rigorously that $$A\leq f(t) \leq B$$ ... ...Note: Although it seems highly plausible, given the definitions of $$A$$ and $$B$$ and given also that $$f$$ is increasing, that $$A\leq f(t) \leq B$$ .. I am unable to prove it rigorously ... Hope someone can help ...
Peter
I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ...
I need some help in understanding the proof of Proposition 3.7 ...Proposition 3.7 and its proof read as follows:View attachment 9509
In the above proof by Andrew Browder we read the following:
" ... ... Clearly $$A\leq f(t) \leq B$$ since $$f$$ is increasing ... ... "
Can someone demonstrate, formally and rigorously that $$A\leq f(t) \leq B$$ ... ...Note: Although it seems highly plausible, given the definitions of $$A$$ and $$B$$ and given also that $$f$$ is increasing, that $$A\leq f(t) \leq B$$ .. I am unable to prove it rigorously ... Hope someone can help ...
Peter