I often encounter reasonings that use the sum of inequalities and this property is intuitive but I was wondering what’s the proof of this operation.
This is what I call the sum of inequalities:
This is the other version, where it’s enough to have a single strict inequality to obtain a strict inequality for the result:
The “proof” is based on two properties of inequalities:
- the transitivity property: for any real number a,b,c
- the addition property: for any real number a,b,c
The “proof” for the non-strict inequality case is:
Given the first two inequalities of the series
we use the addition property with x₁ in the first inequality
we use the addition property with k₁ in the second inequality
we can now use the transitive property on the previous two inequalities (and commutative property of sum) and obtain
The final step is to iterate the procedure for the remaining inequalities.
If all x and k are greater than 0, the “property” holds for the product, as well, as shown in the image below, and also for the strict version of it.
