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I am focused on Chapter 1: Continuity ... ...

I need help with an aspect of the proof of Lemma 1.3.3 ...

Duistermaat and Kolk"s proof of Lemma 1.3.3. reads as follows:View attachment 7681In the above proof we read:

" ... ... The necessity is obvious ... ... "

Presumably this means that if \(\displaystyle \lim_{x \rightarrow a} f(x) = b \) then for every sequence \(\displaystyle (x_k)_{ k \in \mathbb{N} }\) with \(\displaystyle \lim_{k \rightarrow \infty } x_k = a\) we have \(\displaystyle \lim_{k \rightarrow \infty } f( x_k ) = b\) ... ...Although D&K reckon that it is obvious I cannot see how to (rigorously) prove the above statement ...

Can someone please demonstrate a rigorous proof ...

Help will be much appreciated ... ...

Peter