Precise definition of the limit of a sequence

1. Aug 18, 2012

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 $\forall \epsilon > 0$.

2. Aug 18, 2012

garrus

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

3. Aug 18, 2012

micromass

Staff Emeritus
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.

4. Aug 18, 2012

srn

Great, thanks both!