Consider the following two subsets of R S 3 12n n belongst

Consider the following two subsets of R: S: = {3 - 1/2n: n belongsto N} and T:= {q^2: q belongsto Q and q > 0}. Calculate sup 5, inf 5, supT and inf T. Which of the numbers calculated in (a) are a maximum or a minimum?

Solution

S= {5/2, 11/4, 17/6,23/8,....}

Clearly S is increasing set (since an+1> an)

hence infS= 5/2

and sup of S = limit n tends to infinite(3-1/2n) = 3

hence sup(S) = 3

Sup(T) and inf(T) both are does not exists since the upper and lower bounds are not exists