Question

Fill in the blanks to rewrite the following statement with variables: Given any positive real number, there is a positive real number that is smaller. (a) Given any positive real number r, there is s such that __________ s is __________ . (b) For any __________ , __________ such that s < r.

Answer 1

a) there is s such that r>s and s is positive

b) For any r>0 , there exists s>0 such that s<r

Explanation:

a) We are given a positive real number r. We need to wite that there is a positive real number that is smaller. Call that number s. Then r>s (this is equivalent to s<r, s is smaller than r) and s is positive (or s>0 if you prefer). We fill in the blanks using the bold words.

b) The last part claims that s<r, that is, s is smaller than r. We know that this must happen for all posirive real numbers r, that is, for any r>0, there is some positive s such that s<r. In other words, there exists s>0 such that s<r.