Is this definition for partitions in analysis correct?

Question

I think of this new definition -

Suppose $X$ is a bounded interval. A Partition $\mathbb P$ of $X$ is the set $S= \{S_1,S_2,S_3,...,S_n\}$ such that $S_i \subseteq X ~ \forall i$ such that $1\leq i \leq n$ and $\forall ~ x \in X$, $\exists ~j $ such that $x \in S_j$ but $x \notin S_i$ for all $i\ne j$

It first seemed okay to me, but i have a point. When we let $||P|| \to 0$, we know that $n \to \infty$.

And it is quite common in Riemann Integrals to let the normof $P$ tend towards $0$.

Will this cause a problem, like "$n$ becomes infintely large and thus it is not always possible to compare $i$ with $n$" or maybe something else?

Thanks.

Answer 1

Your "new" definition can be rewritten in the common and well known way:

A Partition $\mathbb P$ of $X$ is the set $S= \{S_1,S_2,S_3,...,S_n\}$ such that $$X = \bigcup_{i=1}^n S_i, \quad S_i \cap S_j = \emptyset, i\not= j$$

There are no issues considering $n \to \infty$ because $n$ only tends to infinity but never is. So for each $n \in \Bbb N$ it is finite.