Use of $\mathbb N$ & $\omega$ as index sets

Question

Why all the properties of a sequence or a series or a sequence of functions or a series of functions remain unchanged irrespective of which of $\mathbb N$ & $\omega$ we are using as an index set? Is it because $\mathbb N$ is equivalent to $\omega$?

Answer 1

There is a trivial 1-1 correspondence from $\mathbb N=\{1,2,\dots\}$ to $\omega=\{0,1,2,\dots\}$ given by $f(n)=n-1$. This mapping is a bijection, an order isomorphism, an isometry, and preserves limits, in the sense that $f(n)\to\infty$ if $n\to\infty$ (although it would be hard not to satisfy this). Thus practically every property we care about is preserved when we switch from one index set to the other, almost to the point that we don't need to pay attention to which one we are working with.

Answer 2

It is because $\omega$ and $\mathbb N$ are just different names for the same set. Their members are the same, and so by the Axiom of Extensionality they are the same set.