#Proof

1 messages · Page 1 of 1 (latest)

white warren
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How do I show this. (i) Let $(M,d_M), (N,d_N)$ be metric spaces and $f_n,f:M \rightarrow N$. If $f_n$ converges uniformly against $f$ and for all $n \in \mathbb{N}$, $f_n$ is continues, then show that $f$ is continues as well. Pls explain it to me in every detail. I don‘t know where to start.

past craterBOT
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limber fieldBOT
white warren
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We could start with a $x_0 \in M$

limber fieldBOT
white warren
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And let $\epsilon>0$

limber fieldBOT
dusky merlin
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start with definition of continuity

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what do you have to show for f to be continuous?

white warren
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I have to find a delta>0 s.t $|f(x)-f(x_0)|<\epsilon$

limber fieldBOT
dusky merlin
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don't give half assed answers

white warren
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Kk waitt

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Let $\epsilon>0$ and $x_0 \in M$ . We have to find a $\delta>0$ s.t $d_N(f(x),f(x_0))$, with $0<d_M(x,x_0)<\delta$

dusky merlin
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you're in a metric space

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what does x-x_0 mean?

white warren
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$d_M(x,x_0)$

limber fieldBOT
dusky merlin
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that's better

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  1. By uniform convergence, we can find N such that
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for all n > N

white warren
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Soryy but pls switch $N$ for $n_0$ since $N$ is already used

limber fieldBOT
dusky merlin
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switch it if you want to

white warren
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Can‘t we just make use of the continuety of $f_n$?

limber fieldBOT
white warren
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If we assume

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That

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All f_n are continues

dusky merlin
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that's not enough

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the convergence is also uniform

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you have to make use of both

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by continuity you can also say that

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$$d(f_{n_0}(x),f_{n_0}(z))\leqslant \frac{\varepsilon}{3}$$

limber fieldBOT
dusky merlin
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where d(x,z)< delta

limber fieldBOT
dusky merlin
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now conclude the result by applying the triangle inequality and you are done

white warren
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Why epsilon over 3 tho

dusky merlin
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take epsilon / 11 if you want to

white warren
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How did u choose it

dusky merlin
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doesn't matter

white warren
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Hmm kk

dusky merlin
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same argument but for a real valued function

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adapt to metric space case

white warren
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Okay thanks

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I appreciate your time

white warren
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+close

slender iceBOT
# white warren +close
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