More than convergence

[SemperFi]

I'm trying to find a proof for the following statement:
A sequence b_n attains its limit b, if [FONT=MathJax_Main]∀[/FONT]ε>=0 [FONT=MathJax_Main]∃n₀[/FONT][FONT=MathJax_Main]∀n[/FONT]>=[FONT=MathJax_Main]n[/FONT]₀ ​| b_n-b | <= ε.

How can I prove that the statement is true? Thanks for helping.

 

[HallsofIvy]

What do you mean by "attains its limit"?
What you give is simply the definition that "{b_n} converges to b".

You don't "prove" a definition.