TITLE: Is there any easy proof of Brouwer's fixed point theorem?
QUESTION [7 upvotes]: I am taking a course in game theory. For proving the Nash equilibrium we require  Brouwer's fixed point theorem. But I have not taken a topology course so I am finding the proof difficult to understand. You may explain the same Brouwer's in little easy way.

REPLY [8 votes]: Of course "easy" has a large degree of subjectivity to it.  But there is a famous proof due to Emanuel Sperner that was (and still is) striking for minimizing the amount of topology needed.  
These course notes of Jacob Fox give an exposition using Sperner's Lemma.  The document starts from scratch and proves Brouwer's Theorem (for all $n$) in just under 2.5 pages.