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I am reading John B. Conway's book: A First Course in Analysis and am focused on Chapter 2: Differentiation ... and in particular I am focused on Section 2.1: Limits ...

I need help with an aspect of the proof of Proposition 2.1.2 ...Proposition 2.1.2 and its proof read as follows:

View attachment 9496

In the above proof by Conway we read the following:

" ... ... Now assume that \(\displaystyle f(a_n) \to L\) whenever \(\displaystyle \{ a_n \}\) is a sequence in \(\displaystyle X \) \ \(\displaystyle \{a\}\) that converges to \(\displaystyle a\), and let \(\displaystyle \epsilon \gt 0\). Suppose no \(\displaystyle \delta \gt 0\) can be found can be found tho satisfy the definition. ... ... "

Above Conway seems to me that he is assuming that \(\displaystyle f(a_n) \to L\) and then assuming that the definition of f(a_n) \to L doesn't hold true ... which seems invalid ...

Can someone explain Conway's logic ... can someone please explain what is actually being done in this part of the proof ...

Peter

I need help with an aspect of the proof of Proposition 2.1.2 ...Proposition 2.1.2 and its proof read as follows:

View attachment 9496

In the above proof by Conway we read the following:

" ... ... Now assume that \(\displaystyle f(a_n) \to L\) whenever \(\displaystyle \{ a_n \}\) is a sequence in \(\displaystyle X \) \ \(\displaystyle \{a\}\) that converges to \(\displaystyle a\), and let \(\displaystyle \epsilon \gt 0\). Suppose no \(\displaystyle \delta \gt 0\) can be found can be found tho satisfy the definition. ... ... "

Above Conway seems to me that he is assuming that \(\displaystyle f(a_n) \to L\) and then assuming that the definition of f(a_n) \to L doesn't hold true ... which seems invalid ...

Can someone explain Conway's logic ... can someone please explain what is actually being done in this part of the proof ...

Peter