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If I tell you that: 
sup x ∈ (0,∞)
i.e. 
is the supremum of the set of values for each value of x > 0.
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Then, if this set of values is strictly bounded above by 1/n, why does this imply that the supremum of the set is strictly less than 1/n?
I thought, OK, so let's label the set S = x ∈ (0,∞). Then, we have that all the values in S < 1/n. So, we have that any upper bound of S must be >= 1/n. Hence, any supremum must be smaller than any upper bound, so it must be <= 1/n. So, I had sup(S) = 1/n, and so I wrote sup(S) <= 1/n, and then I used the Sandwich theorem. This was my intuition.
Can someone please help? Thank you
sup x ∈ (0,∞) i.e. is the supremum of the set of values for each value of x > 0. Then, if this set of values is strictly bounded above by 1/n, why does this imply that the supremum of the set is strictly less than 1/n?
I thought, OK, so let's label the set S = x ∈ (0,∞)
. Then, we have that all the values in S < 1/n. So, we have that any upper bound of S must be >= 1/n. Hence, any supremum must be smaller than any upper bound, so it must be <= 1/n. So, I had sup(S) = 1/n, and so I wrote sup(S) <= 1/n, and then I used the Sandwich theorem. This was my intuition.Can someone please help? Thank you
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#2
(Original post by Takeover Season)
If I tell you that: sup x ∈ (0,∞) i.e. is the supremum of the set of values for each value of x > 0.
![Name: Screen Shot 2020-07-19 at 04.14.43.png
Views: 17
Size: 89.1 KB]()
Then, if this set of values is strictly bounded above by 1/n, why does this imply that the supremum of the set is strictly less than 1/n?
If I tell you that:
sup x ∈ (0,∞) i.e. is the supremum of the set of values for each value of x > 0. Then, if this set of values is strictly bounded above by 1/n, why does this imply that the supremum of the set is strictly less than 1/n?
You could actually work out the maximum value of
over the interval 
and it comes to 
, and hence 
I thought, OK, so let's label the set S = x ∈ (0,∞). Then, we have that all the values in S < 1/n.
. Then, we have that all the values in S < 1/n.
seemed fine.
So, we have that any upper bound of S must be >= 1/n.
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(Original post by ghostwalker)
Of itself, it doesn't.
You could actually work out the maximum value of over the interval and it comes to , and hence
Not sure why you called it by another name, S; seemed fine.
No, this isn't true. You can only claim this if you know that 1/n is the supremum (least upper bound). And in fact it isn't. It's just an upper bound.
Of itself, it doesn't.
You could actually work out the maximum value of
over the interval and it comes to , and hence Not sure why you called it by another name, S;
seemed fine.No, this isn't true. You can only claim this if you know that 1/n is the supremum (least upper bound). And in fact it isn't. It's just an upper bound.
over the interval and is the supremum of this set of values. Now, we have shown that each value in the set S .Oh, so now I understand. I can only say that any upper bound of S
either the supremum of S or the maximum of S, and as 
= max/sup(S), I cannot make that conclusion.However, it would've been right instead to say that: any upper bound of S
right?Oh yes, so that means as each value in the set S is bounded above by
, then it must be that 
is an upper bound of S. As the supremum of a set S is less than or equal to every upper bound of a set, then 
.Then, we have
and the conclusion follows.
Last edited by Takeover Season; 5 months ago
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#4
(Original post by Takeover Season)
Hi, thank you for your reply. Sorry, I meant S is the set of values of over the interval and is the supremum of this set of values. Now, we have shown that each value in the set S .
Hi, thank you for your reply. Sorry, I meant S is the set of values of
over the interval and is the supremum of this set of values. Now, we have shown that each value in the set S .
Oh, so now I understand. I can only say that any upper bound of S either the supremum of S or the maximum of S, and as = max/sup(S), I cannot make that conclusion.
However, it would've been right instead to say that: any upper bound of S right?
either the supremum of S or the maximum of S, and as = max/sup(S), I cannot make that conclusion.However, it would've been right instead to say that: any upper bound of S
right?
Oh yes, so that means as each value in the set S is bounded above by , then it must be that is an upper bound of S. As the supremum of a set S is less than or equal to every upper bound of a set, then .
Then, we have and the conclusion follows.
, then it must be that is an upper bound of S. As the supremum of a set S is less than or equal to every upper bound of a set, then .Then, we have
and the conclusion follows.
we can say 
with the strict inequality - in this case. And that justifies its usage in the given text.But without that, or some similiar working, we can only say
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(Original post by ghostwalker)
Sorry, I missed that distinction. My bad.
Yes.
Well we can go a bit further. Knowing that we can say with the strict inequality - in this case. And that justifies its usage in the given text.
But without that, or some similiar working, we can only say
Sorry, I missed that distinction. My bad.
Yes.
Well we can go a bit further. Knowing that
we can say with the strict inequality - in this case. And that justifies its usage in the given text.But without that, or some similiar working, we can only say
Oh right, thank you - that makes perfect sense! Yes, the given working wasn't in the text, so I was confused as to how the strict inequality appeared, but I assume they expect me to do that working out.
Thanks a lot!
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