Suppose that a<0 then choose .
Because of sequence convergence we have
But that means: .
Do you see the contradiction?
I am having a bit of trouble with what should be a simple proof:
Let (an) be a convergent sequence of real numers, an >= 0 for all n in the natural number set. If the limit of an is a (an->a) show that a >= 0.
I understand that I need to use an epsilon neighbourhood to show a contradiction when a is < 0 but am having a hard time finding an epislon that produces a contradiction.
Thanks for the help
This is a hugely important idea in sequence convergence that has a similar application to continuous functions.
If the limit of a sequence is positive then almost all the terms of that sequence must be positive. There is a similar statement of a negative limit.