Consider a cup of coffee. Think of the molecules in that coffee as idealized points in R3, and the set of all molecules in your cup as a compact convex set in R3. Now lift up the cup and gently move it around so that the coffee is stirred in a 'continuous' way, i.e., no splashing! Let the molecules come back to rest. Brouwer's theorem guarantees that at one molecule in your cup is exactly where it was to began with. Reference:Advanced Microeconomic Theory (3rd Edition) by Geoffrey A. Jehle , Philip J. Reny . THEOREM A1.11
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