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## Homework Statement:

- Prove that if f Uniform continuity in R, then for any two series xn and yn limyn-xn=0 so limf(xn)-f(yn)=0

## Relevant Equations:

- Uniform continuity

There are two parts to the question Let's start with part :)

I understand the definition of Uniform continuity And I think I'm in the right direction for the solution but I'm not sure of the formal wording.

So be it ε>0

Given that yn limyn-xn=0 so For all ε>0 , ∃N∈ℕ so that For all N<n , |yn-xn|<ε so ∃δ>0 so that |yn-xn|<δ And because f is Uniform continuity

|f(yn)-f(xn)|<ε.

I know my formulation is not good but I can't formalize it nicely

I understand the definition of Uniform continuity And I think I'm in the right direction for the solution but I'm not sure of the formal wording.

So be it ε>0

Given that yn limyn-xn=0 so For all ε>0 , ∃N∈ℕ so that For all N<n , |yn-xn|<ε so ∃δ>0 so that |yn-xn|<δ And because f is Uniform continuity

|f(yn)-f(xn)|<ε.

I know my formulation is not good but I can't formalize it nicely