Let a be a number with a 1 Prove that a number x is strictl

Let a be a number with a > 1. Prove that a number x is strictly between 1 and a^1/2 if and only if a/x is strictly between a^1/2 and a. (You can assume that 1 < a^1/2 < a).

Solution

Let, 1<x<a^{1/2}

1/sqrt{a}<1/x<1

HEnce,

sqrt{a}<a/x<a

Now,

Let, sqrt{a}<a/x<a

a>1 ,dividing by a

1/sqrt{a}<1/x<1

Hence,

1<x<sqrt{a}