Real Analysis & Topology Summary

GRE Math Subject · Axiom Academy

SUMMARY Real Analysis & Topology Complete unit review: Sequences, continuity, integration, metric spaces, and topological properties Real analysis and point-set topology form the theoretical backbone of the GRE Math Subject Test, appearing in roughly 20-25% of questions. Mastery here means fluency with epsilon-delta arguments, convergence tests, key theorems of calculus, and the topology of metric spaces. A sequence converges to if for every 0"> , there exists such that N |a_n - L| . Key results: Every bounded monotone sequence converges. Every Cauchy sequence in converges (completeness). Bolzano-Weierstrass: every bounded sequence has a convergent subsequence. A series converges if its partial sums converge. Key tests: Comparison Test: If and converges, so does . Ratio Test: . Converges if , diverges if 1"> . Root Test: . Same criteria as ratio test. Alternating Series: If decreases to 0, converges. Integral Test: and converge/diverge together. Absolute vs. conditional: Absolute convergence implies convergence, but not vice versa. converges conditionally. A power series has a radius of convergence where it converges absolutely for and diverges for R"> . Key fact: Within the radius of convergence, power series can be differentiated and integrated term by term. is continuous at if . Equivalently: for every 0"> , there exists 0"> such that . Uniform continuity: The works for ALL points simultaneously, not just one. Continuous functions on compact sets are uniformly continuous.

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