Q:

Let A be a bounded infinite subset of R^2, show that A has at least a limit point. 

Let A be a bounded subset of R^2 with infinite points, show that A has at least one limit point.  
How can a prove that without using compactness?  

A:

enclose $A$ in a square divided into four subsquares.  at least one subsquare contains an infinite number of points.  repeat the process for one such subsquare.  iterate.  the intersection of subsquares converges to a point, which must be a limit point of $A$.