Property of meets and joins (infima and suprema) in a set From a book:

Suppose that $X$ has binary meets and a top element. Then by induction it is easy to see that $X$ has meets of all finite subsets.

So, every 2-element subset of $X$ has a meet (infimum). Why does this imply that every finite subset has a meet too?
A "top element" is just defined to be the infimum of the empty subset, which the author considers finite.
 A: The book is right: roughly speaking, the idea is that you have to compare pairwise the elements of the finite subset of $X$. Let us see that more formally.
Given a finite $Y \subseteq X$, I prove by induction on the cardinality $n$ of $Y$ that $X$ has the meet of $Y$. 
If $n = 0$, then $Y = \emptyset$ and the meet of $Y$ is the top element (which is an element of $X$ by hypothesis).
Let $n > 0$: then, $Y = \{y_1, \dots, y_n\}$ where the $y_i$'s are pairwise distinct. 
By induction hypothesis, $X$ has the meet, say $x$, of $\{y_1, \dots, y_{n-1}\}$. By hypothesis, $X$ has the meet, say $x'$, of $x$ and $y_n$. It is easy to prove that $x'$ is the meet of $Y$.
Indeed:


*

*$x'$ is a lower bound of $Y$, since it is a lower bound of $y_n$ and $x$, and hence of $\{y_1, \dots, y_{n-1}\}$ by transitivity (since $x$ is a lower bound of $\{y_1, \dots, y_{n-1}\}$);

*any lower bound $z$ of $Y$ is a lower bound of $\{y_1, \dots, y_{n-1}\}$ and then $z \leq x$ by maximality of $x$, thus $z \leq x'$ by transitivity; therefore, $x'$ is greater than or equal to any other lower bound of $Y$.
