Difference between $R^\infty$ and $R^\omega$

I know $R^\omega$ is the set of functions from $\omega$ to $R$. I would think $R^\infty$ as the limit of $R^n$, but isn’t that $R^\omega$?
The seem to be used differently, but I can’t tell exactly how.

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In a context where one makes a distiction between $R^\infty$ and $R^\omega$,
$R^\infty$ denotes the set of sequences with finite support whereas $R^\omega$ denotes the set of unrestricted sequences.

In this context, $R^\infty$ is the limit of $R^n$ when $n \to \infty$, in the sense that $R^\infty = \bigcup_{n=0}^{\infty} R^n$, with the convention that $R^n$ are all seen as subsets of $R^\omega$.

The two notations mean exactly the same thing. The second notation is more popular, as many mathematicians are not familiar with ordinal numbers.