Consider the convergent infinite series $\displaystyle S = \sum_{i=0}^\infty a_i$.

If we restrict $a_0, a_1 a_2, \ldots \in \mathbb R$, I think we can all agree that we would have $S \in \mathbb R$. However, this is not true if $a_0, a_1 a_2, \ldots \in \mathbb Q$. Counterexample: $$\sum_{i=0}^\infty \frac{1}{n^2} = \frac{\pi^2}{6} \not\in \mathbb Q$$

My question comes in two parts:
(1) How can we formally prove if $a_0, a_1 a_2, \ldots \in \mathbb R$ then $S \in \mathbb R$?
(2) What distinguishes sets like $\mathbb R, \mathbb C, \mathbb Z$ from a set like $\mathbb Q$ in this sense?

Question

Consider the convergent infinite series $\displaystyle S = \sum_{i=0}^\infty a_i$.

If we restrict $a_0, a_1 a_2, \ldots \in \mathbb R$, I think we can all agree that we would have $S \in \mathbb R$.  However, this is not true if $a_0, a_1 a_2, \ldots \in \mathbb Q$.  Counterexample: $$\sum_{i=0}^\infty \frac{1}{n^2} = \frac{\pi^2}{6} \not\in \mathbb Q$$

My question comes in two parts:
(1) How can we formally prove if $a_0, a_1 a_2, \ldots \in \mathbb R$ then $S \in \mathbb R$?
(2) What distinguishes sets like $\mathbb R, \mathbb C, \mathbb Z$ from a set like $\mathbb Q$ in this sense?

anonymous · Answer

To answer the first question, you'd be showing that the set of real numbers is closed under addition. I believe that's an axiom... If it's not, I have no idea how you'd begin a proof.

I'm not sure how to answer the second question...

anonymous · Answer

It's not quite closure under addition since it's an infinite series.  $\mathbb Q$ is closed under addition, though not all infinite series in $\mathbb Q$ converge to elements in $\mathbb Q$, as demonstrated by my counterexample.