Hey, everyone. I'm trying to prove the following:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] f_n [/tex] and [tex] f_n [/tex] are real-valued function in [tex] \Omega [/tex]

[tex] \{\omega: f_n(\omega) \nrightarrow f(\omega) \} = \\

\bigcup^{\infty}_{k=1} \bigcap^{\infty}_{N=1} \bigcup^{\infty}_{n=1}

\{ \omega : | f_n(\omega) - f(\omega) | \geq 1/k \}

[/tex]

I am convinced by the proof I've made up, but it isn't formal, so I would appreciate if you could help me give it more formality.

Let's call the left side of the equality L and the right side R.

L can be written:

[tex] \exists k \in \mathbb{N} \quad \forall N \quad \exists n \geq N \quad | f_n(\omega) \nrightarrow f(\omega) | \geq 1/k

[/tex]

On the other hand, the last part of R is

[tex] \bigcap^{\infty}_{N=1} \bigcup^{\infty}_{n=1}

\{ \omega : | f_n(\omega) - f(\omega) | \geq 1/k \} [/tex]

which basically takes all the [tex] \omega [/tex] that [tex] \forall N [/tex] have

at least one [tex] n \geq N [/tex] that makes the absolute difference bigger than 1/k

If you take the union for all k, then you have the definition for being in L.

Thanks in advance,

cd

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# Non-convergence written with sets

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