/sci/ - Science & Math

Stupid question about proofs.

It's a common argument in analysis to say: the sequence [math](a_n)[/math] converges to 0, so for any [math]\epsilon>0[/math] we can choose [math]N[/math] so that [math]\sum_{n=N}^{\infty} a_n < \epsilon[/math].

Can I do the same thing when I have a countable set of convergent sequences? That is: the sequences [math](a_{n}^{(i)})[/math] converge to 0 for each [math]i[/math], so we can choose [math]N[/math] so that [math]\sum_{n=N}^\infty a^{(i)} < \epsilon[/math] **for all [math]i[/math]**?

What if there are uncountably many [math](a^{(i)}_n)[/math]?

Not sure if I'm overthinking this.