#Infinite rationals between two reals

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elder steppe
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For 2.3 b) there exists n in N : n(b-a)>1 by archimedian property, so there exists integer m such that for all x
b-a>1/n > |x - m/n| by 2.3 a)
now I was thinking to let x = a once and then x = b to get the desired result, but the m in those two cases would be different right? How do go about solving this?

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granite zephyrBOT
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naive salmon
elder steppe
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so b-a > 1/n is enough?

naive salmon
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Essentially, m/n is in (x - 1/n, x + 1/n)

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Can you pick x and n smartly so that this interval is included in (a, b)?

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Try drawing a picture

elder steppe
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wait let me try

naive salmon
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Sure

elder steppe
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x=a, 1/n = (b-a)/2

naive salmon
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No because m/n can still be outside of (a, b)

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It could be less than x

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You just know it is at distance less than 1/n from it

elder steppe
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oh right

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give me 2 min

naive salmon
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Sure

elder steppe
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x=(a+b)/2, 1/n=(b-a)/2?

naive salmon
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You can't say 1/n = (b-a)/2 but it's a good start

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Take x as the midpoint

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Can you find n such that the conclusion holds?

elder steppe
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2 < (b-a)*n ?

naive salmon
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Like give me a n that fits

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An integer

elder steppe
naive salmon
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This is fine

elder steppe
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what were you expecting though?

naive salmon
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Just that

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Or give me an explicit expression with the floor or something

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Like just prove to me that you can find one such n

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Such that the conclusion holds

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Now you just take this x and n candidates and verify everything fits

elder steppe
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taking x=a+b/2 and 1/n<b-a/2 implies, there exists a rational number between x-1/n, x+1/n which is subset of (a,b) so there's a rational in (a,b)

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and archimedian property gives such a n

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@naive salmon also if you don't mind me asking can i ask about studying math related query?

naive salmon
elder steppe
# naive salmon What's up

so im studying this type of formal math for the first time so should i go ahead and refer a textbook first and then solve these assignments or just try solving these after lectures?

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like my professor just taught lub property, archimedian property and gave this assignment

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I probably dont need to know anything else to solve but its hard and time consuming to find whats the right approach

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like you mentioned about interpreting it as rationals lying at max at 1/n dist, while I was playing around with some inequalities

naive salmon
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I mean I think formalizing is necessary in any case

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regardless of your source material

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But I think you should also find the way to learn that works best for you

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there are people with basically zero geometric intuition

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And it works for them anyway so whatever

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there are people who will also get the problem with just a quick drawing

elder steppe
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hmm probably i also need to draw a mental picture

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also one last thing

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here in (a), isn't it obvious cause S is bounded above or am i missing something?

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cause F = Q or R includes R right?

elder steppe
naive salmon
naive salmon
elder steppe
naive salmon
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The first two sentences argue that S is nonempty

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The remaining argues that there is exists an upper-bound m

elder steppe
naive salmon
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So you're literally quite expected to go into details

elder steppe
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alright thanks a lot

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+thank

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.thank

naive salmon
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close the post, then thank the helper

elder steppe
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+close

coral doveBOT
# elder steppe +close
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naive salmon
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there

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now you can press who to thank

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then close the post

coral doveBOT
# coral dove

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elder steppe
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New system cool

naive salmon
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