#help

74 messages · Page 1 of 1 (latest)

craggy terrace
limpid skiffBOT
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desert cosmos
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thoughts?

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if you see a strict inequality, that indicates the set is not closed

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but double check to be sure

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it's certainly not compact, because it is unbounded. E.g we can take y = n and x=1

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$$ \left\lvert\frac{n}{\sin (n^3)}\right\rvert \geqslant n $$

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@craggy terrace

late thistleBOT
desert cosmos
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heuristically i'd eliminate a,b and c

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so that pretty much leaves d

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verify it's open and you're done

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easiest way I can think of is by showing the complement is closed

craggy terrace
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Can you teach me step by step please?

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sin(n^3) is equivalent to n^3 no?

desert cosmos
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definitely not

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sin is bounded, n^3 is unbounded

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you're thinking of sin x ~ x as x->0, not relevant for my argument

craggy terrace
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I see

desert cosmos
craggy terrace
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Actually no

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But i understands that if it is unbounded then it is open

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Not closed

desert cosmos
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therefore the set is unbounded

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compact sets must be bounded

craggy terrace
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Yes

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I don't understand

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Properly

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🥺🥺

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Rest are fine

desert cosmos
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is the pair (1,10) in the set, for example?

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this is the condition

image.png
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actually I mis spoke

craggy terrace
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,w sin(1000)

late thistleBOT
desert cosmos
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the pair (1,n) is in the set for infinitely many n

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but it still means the set is unbounded

craggy terrace
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Yes it is inside

desert cosmos
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,w sin (8000)

craggy terrace
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10 something

late thistleBOT
craggy terrace
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It still

desert cosmos
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so take (1,20) then

craggy terrace
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Why 20?

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Wait

desert cosmos
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so the result is positive not negative

craggy terrace
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It is still

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But we have mod sign

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So negative can work?

desert cosmos
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this is the condition

image.png
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if sin is negative, that's no good

craggy terrace
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Ohh i got it

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But how we will feel

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Without calculator

craggy terrace
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Sorry a different question

desert cosmos
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i dont remember

desert cosmos
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you should focus your effort in proving the set is open

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rather than all this secondary stuff

craggy terrace
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,w sin(10)

late thistleBOT
craggy terrace
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Sin(5)

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,w sin(5)

late thistleBOT
craggy terrace
desert cosmos
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it is

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sine is positive whenever it is between (0, pi) or (2pi, 3pi) or (4pi,5pi) etc etc

craggy terrace
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So?

craggy terrace
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@desert cosmos

craggy terrace
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+close