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I'm working through Bartle's Elements of Real Analysis, and I'm having trouble parsing Lemma 13.5, on the failure of uniform convergence. The lemma says,

"A sequence $(f_n)$ does not converge uniformly on $D_0$ to $f$ if and only if for some $\varepsilon_0 > 0$ there is a subsequence $(f_{n_k})$ of $(f_n)$ and a sequence $(x_k)$ in $D_0$ such that \begin{align*} |f_{n_k}(x_k) - f(x_k)| \geq \varepsilon_0 \text{ for } k \in \mathrm{N}." \end{align*}

Bartle goes on to say how this can be proved simply by negating the definition of uniform convergence. But I don't understand how the two subsequences, $(f_{n_k})$ and $(x_k)$, arise simply by negating the definition. When I negate the definition, I get something like "there exists an $x \in D_0$ such that, for every $\varepsilon_0 > 0$ and every natural number $K$, there is a corresponding natural number $n \geq K$ for which

\begin{align*} |f_{n}(x) - f(x)| \geq \varepsilon_0." \end{align*}

Any clarification would be appreciated.

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    $\begingroup$ When negate the statement, turn every $\exists$ to $\forall$ and vice versa. In the definition, it is $\forall \varepsilon > 0$, hence in the negated one it should be $\exists \varepsilon_0 > 0$. $\endgroup$
    – xbh
    Oct 1, 2019 at 4:50

1 Answer 1

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$f_n \to f$ uniformly if $\forall \epsilon>0$ exists $N \in \Bbb{N}$ such that $|f_n(x)-f(x)|<\epsilon ,\forall n \in \Bbb{N},\forall x \in D_0$

The negation is:

Exists $s>0$ such that: $\forall N \in \Bbb{N}$ exists $n \geq N$ and $x \in D_0$ such that $|f_n(x)-f(x)| \geq s$

So for $N=1,2,3.... $ exist $n_N \geq N$ and $x_N \in D_0$ such that $|f_{n_N}(x_N)-f(x_N)| \geq s$

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  • $\begingroup$ Yeah, sorry, that was a typo: I know $s$ should merely need to exist. But Bartle's lemma uses subsequences -- that's what I'm having trouble with. Where does that come from? $\endgroup$
    – Rain
    Oct 1, 2019 at 5:07
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    $\begingroup$ @Rain i edited my answer. $\endgroup$ Oct 1, 2019 at 5:12
  • $\begingroup$ Ah, that makes sense -- the subsequence is lurking behind $N$! Thank you. $\endgroup$
    – Rain
    Oct 1, 2019 at 5:14
  • $\begingroup$ You are welcome. $\endgroup$ Oct 1, 2019 at 5:18

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