MAT337 Midterm Summary PDF

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1. The real line State the definitions of supremum and infimum Evaluate the sup/inf of a given set Prove basic facts about sup/inf State the completeness property of R State the Archimedean property. 2. Sequences and series Give the definition of convergence, provide examples of convergent and divergent sequences and prove basic facts about convergence Evaluate limits either straight from the definition or using theorems involving sums/products/quotients of limits. State, prove and use the Monotone Convergence Theorem State, prove and use the Nested Intervals Theorem State and use the Bolzano–Weierstrass Theorem Prove that a Cauchy sequence is the same as a convergent one. Know what an infinite series is and what it means for such a series to (absolutely/conditionally) converge or diverge and be able to provide basic examples. Be familiar with the comparison test and the Leibniz alternating series theorem. 3. The topology of Rn Know the definition of convergence of a sequence in Rn and its equivalence to convergence of each coordinate sequence Find the limits of recursively defined sequences that are Cauchy Define and use the terms closed, complete and compact for subsets of Rn and provide examples of such subsets State and use the Heine–Borel Theorem State and use the Cantor Intersection Theorem...