What is the definition of $\bigcup_{i}^{\infty}A_i$ and $\sum_{i}^{\infty}a_i$?

Question

My assumption until now was $$\bigcup_{i}^{\infty}A_i=\lim_{n\rightarrow\infty}\bigcup_{i}^{n}A_i$$ and $$\sum_{i}^{\infty}a_i=\lim_{n\rightarrow\infty}\sum_{i}^{n}a_i$$

However, I don't recall seeing it anywhere, and I want to verify if my implicit assumptions are correct or not. Thank you for taking the time.

PS: help me find proper tags for this question please.

For $\sum$ you are correct. But for $\bigcup$, a limit of sets doesn't make any more sense than just $\bigcup_i^\infty$ a priori. Read this section for the definition. — Milten, Mar 18 '20 at 14:28

score 0 · Answer 1 · answered Mar 18 '20 at 14:28

0

An element $x$ is in the union iff $x$ is in at least one of the $A_i$. The sum definition is correct.

answered Mar 18 '20 at 14:28

Ray

517

score 0 · Accepted Answer · answered Mar 18 '20 at 14:28

Most of the time, you're right about the infinite sum. There are alternate definitions of infinite sum (like the ones that lead to $\sum_{i=1}^\infty i = -\frac{1}{12}$), but that's probably not of use to you.

As for the union, you don't really need a limit. $\bigcup_{i=1}^\infty A_i$ is the set of all $x$ such that $x \in A_i$ for some $i$.

What is the definition of $\bigcup_{i}^{\infty}A_i$ and $\sum_{i}^{\infty}a_i$?

2 Answers2