Krishna's - Real Analysis, Edition-1D

SYLLABUS- REAL ANALYSIS, Finite and infinite sets, examples of countable and uncountable sets. Real line,
bounded sets, suprema and infima, completeness property of R, Archimedean
property of R, intervals. Concept of cluster points and statement of Bolzano-
Weierstrass theorem.
Real Sequence, Bounded sequence, Cauchy convergence criterion for sequences.
Cauchy's theorem on limits, order preservation and squeeze theorem, monotone
sequences and their convergence (monotone convergence theorem without proof).
Infinite series. Cauchy convergence criterion for series, positive term series, geometric
series, comparison test, convergence of p-series, Root test, Ratio test, alternating
series, Leibnitz's test (Tests of Convergence without proof). Definition and examples
of absolute and conditional convergence. Sequences and series of functions, Pointwise
and uniform convergence. Mn-test, M-test.