- #1
twoflower
- 368
- 0
Hi all,
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
Standa.
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
Standa.
Last edited: