Real Analysis Exam 1 Review Problems and Solutions

https://www.youtube.com/watch?v=EaKLXK4hFFQ. Review of foundational Real Analysis: supremum, Completeness Axiom, limits of sequences, Cauchy sequences, Cauchy convergence criterion, subsequences, Bolzano-Weierstrass Theorem, proofs. https://amzn.to/3GgFjcc ("Real Analysis", by Russell Gordon) 🔴 Real Analysis Course Playlist: https://www.youtube.com/playlist?list=PLmU0FIlJY-MngWPhBDUPelVV3GhDw_mJu 🔴 Abstract Algebra Course Playlist: https://www.youtube.com/playlist?list=PLmU0FIlJY-Mn3Pt-r5zQ_-Ar8mAnBZTf2 🔴 Complex Analysis Course Playlist: https://www.youtube.com/playlist?list=PLmU0FIlJY-MnxAkob30Q5kI0SfPx56Uya #realanalysis #realanalysisreview #realanalysisexam Links and resources =============================== 🔴 Subscribe to Bill Kinney Math: https://www.youtube.com/user/billkinneymath?sub_confirmation=1 🔴 Subscribe to my Math Blog, Infinity is Really Big: https://infinityisreallybig.com/ 🔴 Follow me on Twitter: https://twitter.com/billkinneymath 🔴 Follow me on Instagram: https://www.instagram.com/billkinneymath/ 🔴 You can support me by buying "Infinite Powers, How Calculus Reveals the Secrets of the Universe", by Steven Strogatz, or anything else you want to buy, starting from this link: https://amzn.to/3eXEmuA. 🔴 Check out my artist son Tyler Kinney's website: https://www.tylertkinney.co/ ⏱️TIMESTAMPS⏱️ (0:00) Introduction (0:37) Define supremum of a nonempty set of real numbers that is bounded above (3:43) Completeness Axiom of the real numbers R (5:43) Define convergence of a sequence of real numbers to a real number L (9:06) Negation of convergence definition (11:42) Cauchy sequence definition (13:37) Cauchy convergence criterion (15:39) Bolzano-Weierstrass Theorem (18:58) Density of Q in R (and R - Q in R) (19:37) Cardinality (countable vs uncountable sets) (21:26) Archimedean property (23:40) Subsequences, limsup, and liminf (28:43) Prove sup(a,b) = b (35:34) Prove a finite set of real numbers contains its supremum (39:39) Find the limit of a bounded monotone increasing recursively defined sequence (45:28) Prove the limit of the sum of two convergent sequences is the sum of their limits (51:28) Use completeness to prove a monotone decreasing sequence that is bounded below converges (58:14) Prove {8n/(4n+3)} is a Cauchy sequence AMAZON ASSOCIATE As an Amazon Associate I earn from qualifying purchases.