Weakly Summaries

Week 1: We started the study of the real numbers. The material we went over corresponds, in the notes, to Section 1 up to the formulation (but not the proof) of Lemma 1.7. 

Details: After reviewing the axioms of the real numbers pertaining to arithmetic and order, we saw that the rational numbers do not contain a number whose square is two.
Then we introduced the notions of upper bound and supremum of a set of real numbers and saw the formulation of the least upper bound axiom. Finally, we saw how the LUB axiom implies that there exists a real number whose square equals two, and we showed that the set of natural numbers is not bounded above (the Archimedean property).

Week 2: We started the study of sequences. The material we went over corresponds, in the notes, to the end of Section 1 (including the proof of Lemma 1.7 and everything after it) and the first part of Section 2 (up to the end of the proof of Lemma 2.9). As an exception, we did not cover the proof of the triangle inequality (Lemma 1.9 in the notes) and, as always, none of the sections with a * are covered (eg. Sections 1.4 and 1.7).

Details: We proved the interlacing property (between any two rationals there is an irrational and vice-versa), introduced the triangle inequality and started the study of sequences, including the definition of convergent sequence and some examples.
We proved the uniqueness of the limit and the "shift rule". We saw some examples of convergent sequences. We defined bounded sequences, showed that every convergent sequence is bounded and then established the algebra of limits (the "sum rule", the "product rule" and the "quotient rule").

Week 3: We continued the study of sequences and their limits. This week we finished Section 2 in the notes.

Details: We saw some applications of the algebra of limits, then studied interactions between limits and inequalities, including the "sandwich rule". We saw examples of application of the sandwich rule, then introduced infinite limits and proved some properties. We finished with the "ratio test for limits" and saw several applications.

Week 4: We continued the study of sequences and their limits. In the notes this corresponds to Section 3.

Details: We introduced monotonic (i.e. increasing or decreasing) sequences, saw some examples, then proved that every increasing and bounded sequence converges and used this result in some examples. We saw the proof that the sequence (1+1/n)^n is convergent. Then we introduced the concept of subsequences and stated and proved the Bolzano-Weierstrass Theorem. We introduced the notion of Cauchy sequences; proved that a sequence is convergent if and only if it is Cauchy and then discussed decimal expansions of real numbers as an example of convergent sequences.

Week 5: We started the study of series. We covered Section 4 in the notes up to (and including) Subsection 4.5. We did not cover sections 4.3.1, 4.3.2, 4.4.

Details: Introduced infinite series, defined convergent series, saw some examples and some basic properties of convergent series (including the vanishing of the tail). Explored series with non-negative terms and saw several examples (the harmonic series, the Basel problem and the definition of the number e). We revisited the comparison test and saw some more examples. Then we introduced the notion of Absolute convergence and saw how it implies regular convergence.

Week 6: We continued to study infinite series. We covered the rest of Section 4 in the notes, starting at subsection 4.6 (we did not cover 4.7.2 in detail and only saw the statement of Lemma 4.15).

Details: Ratio test, root test and integral test with examples. Applications of the integral test to the analysis of the series ∑1/n^p
 for 1<p<2. Alternating series test. Rearrangements of series. We saw how convergent series with positive terms can be rearranged and always give the same value, then we saw that absolutely convergent series also have this property, and finally we proved that a conditionally convergent series can be rearranged to equal any real number.

Week 7: We begin the study of real valued functions of one real variable (i.e. functions f:E→R where E⊂R). In the notes this week corresponds to Section 5.

Details: Introduced some notation and terminology for functions and saw several examples. Defined continuity of a function at a point, defined the auxiliary notion of δ-neighborhood of a point within a domain and saw some examples of continuous functions. Saw some examples of continuity, and basic properties of continuous functions, including the preservation of sign and the algebra of continuous functions. Introduced the notion of composition of functions and saw that the composition of two continuous functions is continuous. Finally, we derived some corollaries, including the fact that the ratio of two continuous functions is continuous.

Week 8: We established a bridge between continuity of functions of a real variable and convergence of sequences. This corresponds to Chapter 6 in the notes. We also saw an example from Chapter 8, but most of that chapter was not covered (only the proof that sin x is continuous).

Details: We proved that a function is continuous if and only if it is sequentially continuous and saw some examples, including trigonometric functions.

Week 9: More advanced properties of continuous functions, including the Intermediate Value Theorem and the Extreme Value Theorem. This covered essentially all of Chapter 9 in the notes, with the exception of the proof of Lemma 9.7, which characterizes intervals.

Details: Intermediate Value Theorem with full proof and applications, including the fact that every odd degree polynomial has a root in R, and that the image of an interval under a continuous function is again an interval. Extreme Value Theorem, including a full proof. As a corollary we obtained the fact that the image of a closed interval under a continuous function is again a closed interval.

 Week 10: Explored the notion of uniform continuity and saw the inverse function theorem. In the notes this corresponds to Chapters 10 and 11.

Details: Uniform continuity - definition and examples. Theorem 10.1 which states that a continuous function on a closed interval [a,b] is uniformly continuous. Finally, we formulated and proved the Inverse Function Theorem (Theorem 11.2).