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1-1 mapping absolutely convergent arbitrary Bolzano-Weierstrass theorem bounded function bounded open set bounded set called Cantor set cardinal number Cauchy sequence Chapter closed set completes the proof complex numbers consider construct Corollary countable number Definition denoted dense denumerable derivative divergent element equivalent example Exercise exists finite measure finite number fn(x following theorem Fourier series Heine-Borel theorem Hence Hint implies inequality infinite interior point intersection last theorem Lebesgue integrable lim inf lim sup limit point linear order me(S measurable function measurable sets natural number neighborhood nondenumerable nonempty set nonnegative number system open ball open interval open set ordinal pairwise disjoint partition path-connected point x0 polygonal path proof is complete proof is left prove the following Prove Theorem rational numbers real numbers Riemann integral semicontinuous set of type Show subset summable unbounded uniformly convergent upper bound well-ordered set write zero