- #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?