CIJS,
I've completed my self-study of real analysis. The subjects that I covered during this fascinating intellectual odyssey were the properties of real numbers, basic topology of the real numbers, sequences, series, functional limits, continuity, differentiation, and Riemann integration. The final topic that I studied and was actually able to appreciate and understand via relevant definitions, propositions, theorems, mathematical ideas, etc., was the fundamental theorem of calculus (FTC). I can't believe that the theoretical beauty of this wonderful idea had eluded me when I first studied calculus: my computational mindset inhibited my ability to appreciate the harmony of mathematical concepts and ideas that make the FTC work.
That said, I must say that learning real analysis has given me a much greater appreciation for the originality, creativity, perseverance, and wonder that is involved in successfully constructing convincing proofs that establish the truth of mathematical statements; I'm truly in awe of those individuals who are gifted at proof-based mathematics.
I've completed my self-study of real analysis. The subjects that I covered during this fascinating intellectual odyssey were the properties of real numbers, basic topology of the real numbers, sequences, series, functional limits, continuity, differentiation, and Riemann integration. The final topic that I studied and was actually able to appreciate and understand via relevant definitions, propositions, theorems, mathematical ideas, etc., was the fundamental theorem of calculus (FTC). I can't believe that the theoretical beauty of this wonderful idea had eluded me when I first studied calculus: my computational mindset inhibited my ability to appreciate the harmony of mathematical concepts and ideas that make the FTC work.
That said, I must say that learning real analysis has given me a much greater appreciation for the originality, creativity, perseverance, and wonder that is involved in successfully constructing convincing proofs that establish the truth of mathematical statements; I'm truly in awe of those individuals who are gifted at proof-based mathematics.
