Definition of a finite set of points in R What is a finite set of points in $\mathbb{R}$?
I am trying to prove by induction that any finite set of points in $\mathbb{R}$ is compact.
 A: Luckily, you don't need to worry about whether "finite" means "Dedekind-finite" (as in Tushant's answer, that is $\forall x\subsetneq A,|x|<|A|$) or "equinumerous to $\{1,\dots,n\}$ for some $n$" or something, because "finite" appears explicitly in the definition of "compact".
Most importantly for your induction proof, finite sets are something that you can do induction on: if property $P$ holds of the empty set and it holds for $x\cup\{y\}$ assuming it holds for $x$, then property $P$ holds for every finite set. And conversely, the empty set is finite and $x\cup\{y\}$ is finite when $x$ is. These two properties uniquely characterize finite sets, similar to the Peano axioms for the natural numbers, if you are familiar with that. (It turns out that this property corresponds with the second definition above; Dedekind-finiteness has some fiddly properties without some form of the axiom of choice.)
For our property $P$ we take $P(x)$ to be "$x\subseteq\Bbb R\implies x$ is compact". The empty set is compact because any open cover of $\emptyset$ has a finite subcover, namely $\emptyset$. And assuming that $x$ is compact, given an open cover $U$ of $x\cup\{y\}$ there is some $U_y\in U$ with $y\in U$, and some finite cover $\{U_i\}_{i\in I}$ of $x$, so $\{U_i\}_{i\in I}\cup\{U_y\}$ is a finite cover of $x\cup\{y\}$.
A: A definition of a finite set could be that:   a set S is finite iff for every proper subset G,
$$|G|<|S|$$.
Another one that I can think of that is ,
a set S is finite iff there exist $$f:S \to N$$ and a natural number number n such that $f(s)<n \   \forall\  \ s \in S$ 
A: A finite set of points has a finite set of subsets. If there are $n$ points, there are only $2^n$ subsets.  Any cover will be a subset of those $2^n$, so will be finite and will be the finite subcover you are looking for.
