Precise definition of the limit of a sequence

[srn]

In the definition,

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

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

 

[garrus]

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

 

[micromass]

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

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

 

[srn]

Great, thanks both!

 

[blue_raver22]

1) The reason we must find a n_0 \in N such that \forall N \geq n_0 is because the limit of a sequence is defined as the value that the terms of the sequence approach as n (the index of the term) becomes larger and larger. By choosing a n_0 \in N, we are ensuring that the terms of the sequence are approaching a specific value as n gets larger, rather than just approaching any value in the real numbers. This helps to provide a more precise and specific definition of the limit.

2) The inequality |x_n - a| < \epsilon must hold because it ensures that the terms of the sequence are getting closer and closer to the limit value, a. If we were to use |x_n - a| \leq \epsilon, it would allow for the possibility of the sequence oscillating around the limit value, rather than approaching it. By using the strict inequality, we are ensuring that the terms are getting arbitrarily close to the limit value as n gets larger, rather than just being within a certain range of it. This is important in defining the limit of a sequence as it captures the idea of approaching a specific value, rather than just being within a certain distance from it.