- #1

Math Amateur

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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 \(\displaystyle A\leq f(t) \leq B\) since \(\displaystyle f\) is increasing ... ... "

Can someone demonstrate, formally and rigorously that \(\displaystyle A\leq f(t) \leq B\) ... ...Note: Although it seems highly plausible, given the definitions of \(\displaystyle A\) and \(\displaystyle B\) and given also that \(\displaystyle f\) is increasing, that \(\displaystyle A\leq f(t) \leq B\) .. 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 \(\displaystyle A\leq f(t) \leq B\) since \(\displaystyle f\) is increasing ... ... "

Can someone demonstrate, formally and rigorously that \(\displaystyle A\leq f(t) \leq B\) ... ...Note: Although it seems highly plausible, given the definitions of \(\displaystyle A\) and \(\displaystyle B\) and given also that \(\displaystyle f\) is increasing, that \(\displaystyle A\leq f(t) \leq B\) .. I am unable to prove it rigorously ... Hope someone can help ...

Peter