- #1
sergey_le
- 77
- 15
- 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