the meaning of "finite" in the finite covering theorem A textbook I am using to learn analysis states (in reference to just the real line):

Every system of open intervals covering a closed interval contains a finite subsystem that covers the closed interval.

(the textbook is "Mathematical Analysis I" by V.A. Zorich)
Let's say $S$ = {$U_n$} is the system of open intervals $U_n$ in question, which covers the closed interval $I$.
Now, if we take all $U_n$ to be infinitely small in length, but take $S$ to be infinite in cardinality such that all $U_n$ still cover $I$ (does this even make sense?), wouldn't it be impossible to select a finite subsystem of $S$ that covers $I$?
I feel that maybe I am missing an elementary but important distinction in my idea of "infinite"

Your mistake lies in the assumption that you can choose "$U_n$ infinitely small in length". For each $n$ seperately the set $U_n$ is an open set. An open set contains an open interval and an open interval has a positive measure (the notion of "length" doesn't really make sense for an arbitrary open set, but that is not the problem here). Obviously the open sets may become "smaller" (again, you have to be careful how you measure the size of an open set) if $n$ varies, but that's a different story.
Btw: The property you quote is the definition of compactness. For subsets of the real numbers we have: bounded and closed if and only if compact. Note that the textbook presumably assumes that the interval is bounded.
