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- Jun 22, 2012

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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:

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