- #1

danago

Gold Member

- 1,123

- 4

[tex]

\begin{array}{l}

A = Q \cap (\sqrt 2 ,\infty ) \\

B = \{ n + \sin n|n \in Z^ + \} \\

C = \{ 0.1,0.01,0.001...\} \\

\end{array}

[/tex]

From the definition of supremum, it is obvious that sup A does not exist, because for any rational number x/y in set A, (x+1)/y is also in A and is greater than x/y, hence there are no upper bounds.

Im a bit stuck with the infimum of A. I know that it exists because 0 is a lower bound of A, therefore an infumum must exist. I would be inclined to say that it is just sqrt(2), but not really sure how to justify it. Is it true that in every interval, there exists a rational number? If this is the case (which i believe it is), then for any rational number m, there will be some other rational number within the interval (sqrt(2), m), and hence m cannot be the lowest upper bound. Is this reasoning valid?

For set B, sup B will not exist because for any element x + sin x, (x+5) + sin (x+5) is greater and also within the set, hence no upper bound exists (since the sine function has a range small finite range), and inf B will just be 1+sin 1?

I had no troubles with C. I just said that sup C=0.1 and inf C=0.

Does that look right? Its mainly the first set i wasnt sure about, mainly the justification.