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Prove that the square root of 2 is irrational.
Prove the Cauchy-Schwarz inequality: for real numbers a_1,...,a_n and b_1,...,b_n, we have (sum a_i*b_i)^2 <= (sum a_i^2)*(sum b_i^2).
Prove that every continuous function from R to R is differentiable.
Prove Fermat's Little Theorem: for any prime p and integer a not divisible by p, a^(p-1) is congruent to 1 modulo p.
Prove that every finite integral domain is a field.
Let a, b, c be positive real numbers such that abc = 1. Prove that a + b + c >= 3.