# Answer to Question #3401 in Real Analysis for junel

Question #3401

If 0 < a < b, show that

1)& a < (ab)

2) 1/b < 1/a.

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

thus

a

Since a, b are positive, √(a

a < (ab)

In similar way ( * b) we can obtain (ab)

a < (ab)

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.

Let's multiply the given inequality:

a < b | x a

thus

a

^{2}< abSince a, b are positive, √(a

^{2}) < √ (ab)a < (ab)

^{1/2}In similar way ( * b) we can obtain (ab)

^{1/2}< b, thusa < (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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