Introduction to Real Variable Theory |
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a₁ a₂ absolutely convergent an+1 b₁ Bolzano-Weierstrass theorem bounded function bounded open set bounded set called Cantor set cardinal number Cauchy sequence closed interval closed set completes the proof complex numbers contains Corollary countable Definition denoted dense denumerable divergent element example Exercise exists f(xo F₁ F₂ finite number fn(x following theorem Fourier series function defined function f ƒ is continuous G₁ G₂ Hence Hint implies inequality interval a,b L₁ L₂ Lebesgue integral Let f Let f(x lim f(x lim inf lim sup limit point linear order lower bound measurable function measurable sets metric space natural number neighborhood nondenumerable nonnegative open interval open set ordinal pairwise disjoint partition path-connected proof is complete Prove Theorem rational numbers real numbers result Riemann integral S₁ S₂ Show subset summable type F U₁ U₂ unbounded uniformly convergent zero