We started class today by presenting challenge problems. We reviewed ways to solve problems 1a,b,c, 2 a,b, and 3 a,b. We also proved several theorems today. Theorem 2.3.2 says that every convergent sequence is bounded. Using the definition of convergence, we are able to prove this theorem in only a few lines.

Theorem 2.3.3 is the Algebraic Limit Theorem, which has four parts. Part one says that the limit of a constant times a sequence converges to the constant times the limit of the sequence. Part two says that the limit of the sum of two sequences converges to the sum of the two limits of the sequences. We did not prove this part in class, but it is a good exercise to do on your own. Part three says that the limit of the product of two sequences converges to the product of the two limits of the sequences. Part four says that the limit of the quotient of two sequences converges to the quotient of the two limits of the sequences if the limit in the denominator does not equal zero. These proofs all use epsilon arguments. For part one and three, we need to approach the proofs with two cases. One case is the zero case, which is usually the easier case to argue. The second case is the non-zero case, which takes more work but is still manageable. It is sufficient to show that the limit of converges to in order to prove part four because a quotient is the product of and .

We ended class by stating the Monotone Convergence Theorem (2.4.2). We did not prove it in class, but we did define . This led to our end of class discussion about why do we know that exists. We determined that exists because if we negate the set , that set will have a . Thus, the negation of the negation of (just ) has a .