# Limits of Functions ...Conway, Proposition 2.1.2 ... ...

#### Peter

##### Well-known member
MHB Site Helper
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:

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