Precise definition of the limit of a sequence

[srn]

In the definition,

1) why must you find a n_0 \in N such that \forall N \geq n_0? You might as well say find a n_0 \in R such that \forall N > n_0. Just a matter of simplicity?

2) Why must |x_n - a| < \epsilon hold? I think |x_n - a| \leq \epsilon is fine as well, given that it must hold \forall \epsilon > 0.

 

[garrus]

1)They are subscripted by natural numbers in general ,i presume for simplicity and countability.

 

[micromass]

1) Yes, taking n_0\in \mathbb{R} works as well. But it is often simper to take n_0\in \mathbb{N}.

2) Having <ε or ≤ε makes no difference. Both definitions work and are equivalent.

 

[srn]

Great, thanks both!