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I need some help about topology homework.. anyone..

Mathematics
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first one i can do negate the definition by picking \(\epsilon =\frac{1}{2}\) then show that there does not exist a \(\delta\) such that \(|x|<\delta\implies \sin(\frac{1}{x})<\frac{1}{2}\)

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that is because no matter how small \(\delta\) is there will be some \(k\) such that \[\frac{2}{(2k+1)\pi}<\delta\] and for that \(k\) and all other larger you will have \(\sin(\frac{1}{x})=1\)
yes I believe we can choose epsilon, since it is negation..
thanks it makes sense to me..
sin(1/x) must be greater than epsilon, and it is true, how about 1/(2kpi)
@satellite73 hi, satellite73, can you check my comment here
hi do you know exactly how to negate the definition of \[\lim_{x\to a}f(x)=L ?\]that is what you need to use
definition of \[\lim_{x\to a}f(x)=L \] \(\forall \epsilon> 0\), \(\exists \delta \) such that \(|x-a|\implies |f(x)-L|<\epsilon\)
yes..
it should be I guess \[|x-a|\implies |f(x)-L|\ge \epsilon\]
negation is \(\exists \epsilon >0\) such that \(\forall \delta\), \(\exists x\) such that \(|x-a|<\delta\) and \(|f(x)-L|>\epsilon\)
yes this is what I know..
you need to use what i wrote above (took me a while) so the logic is this you exhibit a specific \(\epsilon\) and then show that no matter how small \(\delta\) is, then you can find some \(x\) for which \(|x-a|<\delta\) and at the same time for that \(x\) you have \(|f(x)-L|>\epsilon\) my suggestion was to pick \(\epsilon =\frac{1}{2}\) whatever you pick, it should be specific not general
in your example, you have to show that \(\exists \epsilon>0\) so that \(\forall \delta\) \(\exists x\) with \(|x|<\delta\) and \(|\sin(\frac{1}{x})|>\epsilon\)
yes ..
but it is gonna be contradiction right, and we are gonna say f(x) is not cont. at x=0..
since \(\sin(\frac{1}{x})=1\) infinitely often close to zero, you can show this pick \(\epsilon=\frac{1}{2}\) for example, then \(\forall \delta>0\) \(\exists \frac{1}{(2k+1)\pi}<\delta\) and so \(\sin(\frac{(2k+1)\pi}{2})=1>\frac{1}{2}\)
you are negating the definition, but this is not a proof by contradiction, it is a direct proof, using the negation of the definition
a proof by contradiction would be to assume it is continuous and then arrive at some contradiction, but you do not need to do this here. write down exactly what the negation of the definition is, then prove it directly
can I use my example to get contradiction..sin(1/x)=0 but greater than 1/2
not greater than.
i should have said \(\forall \delta>0\) \(\exists k\) such that \(\frac{1}{(2k+1)\pi}<\delta\)
it makes more sense now..
be careful here, you cannot write \[\sin(\frac{1}{x})=0\] that makes no sense
again, i stress work the directly from the definition of " \(f\) is not continuous at \(a\)" i know it seems like a proof by contradiction, but it is not the definition of "\(f\) is not continuous at \(a\)" i wrote above
thanks.. it must be grater than 0
question 2 looks annoying but i think you can do it by saying something like this for any two points you can join them by a straight line (you can even write down what the line would be) unless the line would contain those two points, in which case you can join them by a horizontal and vertical line that skips those points probably an easier way, not sure
yeah, I thought something like that.. how about 5)ii)
for 5) ii) can we say that since A is compact it is finite.. I know it is bounded and closed.. is it true that every closed and bounded set is finite..

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