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Answer to Question #3401 in Real Analysis for junel

Question #3401
If 0 < a < b, show that
1)& a < (ab)1/2 < b;
2) 1/b < 1/a.
Expert's answer
0 < a < b
Let's multiply the given inequality:
a < b | x a
a2 < ab
Since a, b are positive, √(a2) < √ (ab)
a < (ab)1/2
In similar way ( * b) we can obtain (ab)1/2 < b, thus
a < (ab)1/2 < b.
There are positive numbers a and b, such that a < b. Let's divide this inequality by a:
a/a < b/a.
1 < b/a.
Then divide it by b and get:
1/b < 1/a. The statement is proved.

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