X1. Prove that if 0 < c < 1, then the sequence {cn} converges to 0.
X2. Prove Lemma 2.42. The book's proof is missing a lot. Prove every step rigorously. In the last sentence of the proof, I believe "majorizes" means "is greater than or equal to." You may use the fact that for every x > 0, x0 = 1 (this is by definition).
Hint: Prove the last sentence of the proof using the definition of alpha^beta given right before the lemma.
X3. Prove or disprove: If a_n is a bounded sequence and b_n diverges to infinity, then a_n / b_n converges to 0.
X4. Let a_n be a sequence of real numbers, and e any positive real number. Prove that if lim sup a_n = S is finite, then for all sufficiently large n, a_n < S + e. Give a corresponding statement for lim inf (without proof).