Problem Statement: Let be a compact subset of and a continuous function. Show there exists a such that for every .
Proof: Since is compact and is continuous it follows that is compact. Bolzano-Weierstrass gives us that is closed and bounded. This means there is both an upper and a lower bound for . Since is continuous it follows that there is a such that is a minimum since is continuous on a compact set. Let . Then is a lower bound for . We know that since each which means that the minimum value of must be strictly greater than .
Thus, there exists a such that for every .
Reflection: The key for this problem is that continuous functions map compact sets to compact sets. Then we know that the function attains it’s minimum value since is a continuous function on a compact set. Since we were given that for every we know that this minimum value is greater than .