Week 5-content
We finished the proof of the nested intervals lemma (Rudin, Theorem 2.38, p. 38). We applied it to show that every bounded sequence of real numbers has at least one accumulation point (equivalent to the case of Theorem 3.6(b) when k=1, p. 51–52).
We defined convergent sequences and their limits (Definition 3.1, p. 47). (You can treat X=R, d(x,y)=|x-y| and "points" to be real numbers in Rudin's notation) We proved some properties (throughout 3), including the monotone convergence theorem (Theorem 3.14, p. 55) and characterization of convergent sequences as those that are bounded and have precisely one accumulation point (not stated as a separate theorem in Rudin, but could be deduced as consequence of 3.2(c), p. 48, 3.18(c), p. 57, but it uses liminf and limsup, which we cover next week).
We studied operations on sequences and the effects on their limits (Theorem 3.3, p. 49–50). We also introduced divergence to + infinity and - infinity (Definition 3.15, p. 55–56).