- #1

Kernul

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## Homework Statement

##I## a set of real number and ##f_k : I \rightarrow \mathbb{R}## a succession of real functions defined in ##I##.

We say that ##f_k## converges punctually in ##I## to the function ##f : I \rightarrow \mathbb{R}## if

$$\lim_{k \to \infty} f_k(x) = f(x), \hspace{1cm} \forall x \in I$$

Which is like saying:

$$\forall \epsilon > 0,\hspace{1mm} \forall x \in I,\hspace{1mm} \exists \nu_{\epsilon, x} \in \mathbb{R} : |f_k(x) - f(x)| < \epsilon \hspace{1cm} \forall k > \nu_{\epsilon, x}$$

## Homework Equations

## The Attempt at a Solution

What is the ##\nu_{\epsilon, x}##? Is it the same in the normal succession of real numbers? Could someone make some examples?

And I've read that for the uniform convergence, this ##\nu_{\epsilon, x}## depends only from ##\epsilon##, so ##\nu_{\epsilon}##, and that the uniform convergence implies the punctual convergence but not vice versa. Any example with these too?