Find the supremum and infimum of S, where S is the set
S = {√n − [√n] : n belongs to N} .
Justify your claims. (Recall that if x belongs to R, then [x] := n where n is the largest integer less than or equal to x. For example, [7.6] = 7 and [8] = 8)
I found my infimum to be 0 and my supremum to be 1, but how do i go about proving them? Help please.
I took a different approach, not sure if its right:
If 1 is an upper bound of S that satisfies the stated condition and if v < 1, then ε = 1-v. Since ε > 0, there exists Sε ∈ S, such that v = 1- ε < Sε, therefore v is not an upper bound of S, and we conclude that 1 = sup S
and for the infimum:
If 0 is a lower bound of S and if t > 0, then ε = t-0. Since ε > 0, there exists Sε ∈ S, such that t = ε - 0 > Sε. Therefore, t is not a lower bound of S, and we conclude that 0 = inf S
Is this right?
What about if i prove the infimum as follows:
By definition of [x], x≥[x] so the terms of the sequence are always positive thus 0 is indeed a lower bound.
Now suppose a was a lower bound and a>0. Taking n=4 we see that a cannot be a lower bound since a>0 . Thus our assumption that a is a lower bound is false. So inf S = 0.
and for the supremum, i do sort of the same:
By definition of {x}, {x} < 1 since x - [x] = {x}, therefore {x} never reaches 1. So 1 is an upper bound.
Let b be an upper bound where b < 1. If you take ?, (what value would i be able to use here to prove this, since 1 isnt part of the set) or this where are use the limit you displayed above?