Answer to Question #11860 in Real Analysis for sneha tambe

Question

let S be a non-empty bounded set in real numbers a) let a>0 , and let aS:={as; s belongs to S} prove that inf(aS)=a inf S, sup(aS)= a sup S

Answer 1

Let s belongs to S, so
inf(S) <= s <= sup(S),
whence

a*inf(S) <= a*s <= a*sup(S),
and therefore
a*inf(S) <=
inf(aS) and sup(aS) <= a*sup(S).

We should prove the inverse
inequalities:

a*inf(S) >= inf(aS) and sup(aS) >=
a*sup(S).



1) Suppose a*inf(S) < inf(aS), so

inf(S)
< inf(aS) / a

Therefore there exists s from S such that


inf(S) <= s < inf(aS) / a,

whence

(*) a*s <
inf(aS).

But a*s belongs to aS, and so the inequality (*) is
impossible.
Hence a*inf(S) = inf(aS).



2) The proof that
sup(aS) >= a*sup(S) is similar.

Suppose sup(aS) < a*sup(S),
so

sup(aS)/a < sup(S)

Therefore there exists s from S such
that

sup(aS)/a < s <= sup(S)

whence

(**)
sup(aS) < a*s.

But a*s belongs to aS, and so the inequality (**) is
impossible.
Hence sup(aS) = a*sup(S).