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## Homework Statement

Prove that if 0 < a < b,

then a < √ab < (a + b)/2 < b

## Homework Equations

To prove this use the 12 properties of numbers (commutativity, trichotomy law, associativity, etc...).

## The Attempt at a Solution

The main problem is I don't know if I need to define the square root function since I'm doing a proof. I'm new to all this, this is my first time doing analysis, so bear with me.

I assume you break up the inequality into parts, since a<b is already defined, we don't need to prove the whole inequality. So I did the first part, unintutively

a<√ab

a^2 < (ab)

(a)(a) (a^-1) < (ab)(a^-1)

a<b which was assumed thus the inequality is proven

Also

(a + b)/2 < b

(a +b)(2^-1) (2) < b(2)

a < b

Problem is I don't know what to write division of 2 as, since it's not listed in basic property. Also I don't know how to prove the 2 centre terms of the inequality: √ab < (a + b)/2