Hi all,(adsbygoogle = window.adsbygoogle || []).push({});

I'm learning some calculus theory and I found one point I don't fully understand:

[tex]

\mbox{Let M} \subset \mathbb{R} \mbox{ be non-empty set and let } f, f_{n}, n \in \mathbb{N} \mbox{ be functions defined on M. Then the following is true:}

[/tex]

[tex]

f_n \rightrightarrows f \mbox{ on M} \Leftrightarrow \lim_{n \rightarrow \infty} \sup \left\{\left|f_{n}(x) - f(x)\right|; x \in M \right\} = 0

[/tex]

Proof:

[tex]

f_{n} \rightrightarrows f \mbox{ on M }

[/tex]

[tex]

\Leftrightarrow\ \forall \epsilon > 0\ \exists n_{0} \in \mathbb{N}\ \forall n \geq n_{0} \forall x \in M\ : |f_{n}(x) - f(x)| < \epsilon

[/tex]

[tex]

\Leftrightarrow \forall \epsilon > 0\ \exists n_{0} \in \mathbb{N}\ \forall n \geq n_{0} : \sup_{x \in M} \left\{ | f_{n}(x) - f(x) | \leq \epsilon \right\}

[/tex]

[tex]

\Leftrightarrow \lim_{n \rightarrow \infty} \left( \sup_{x \in M} |f_{n}(x) - f(x)| \right) = 0

[/tex]

I don't get why in the last but one condition in the proof there is [itex]\leq \epsilon[/tex] instead of [itex]< \epsilon[/tex].

Could you please tell me the reason?

Thank you very much

Best regards,

Standa.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Equivalent characterization of uniform convergence

**Physics Forums | Science Articles, Homework Help, Discussion**