Prove a set can have at most one greatest lower boundSolutio

Prove a set can have at most one greatest lower bound

Solution

Answer :

let S be a set of real numbers and u₁ and u₂ be greast lower bounds for S.

We have to prove that u₁ = u₂

So, since u₁ is a greast lower bound for S and u₂ is an lower bound for S we have u₂ u₁.

Similarly, since u₂ is a greast lower bound for S and u₁ is an lower bound for S we have u₁ u₂ .

But u₁ u₂ and u₂ u₁ implies that u₁ = u_2.

Thus a set can have at most one greatest lower bound