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chapter page 1 Sets
Sets of real numbers
Countable and uncountable sets
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Amer Baire's theorem Bolzano-Weierstrass theorem boundary points Cantor set chord of length closed sets closure consider construct contains a neighborhood continuous function contradiction converges uniformly convex countable number countable set covered decimal defined definition dense set diam diameter differentiable digits discontinuous disjoint distance elements empty everywhere dense set example Exercise exists finite derivative finite number given graph Heine-Borel theorem Hence horizontal chord inequality interior point intermediate value property intersection interval 0,1 inverse image least upper bound lim sup limit point lower bound Math maximum mean-value theorem measure zero metric space monotonic function nondecreasing nonempty set number of points one-to-one correspondence open intervals open sets ordered pairs partial sums pointwise polynomial positive integers positive number proof prove put into one-to-one rational numbers rational points real numbers right-hand set of measure set of points Similarly subinterval subset suppose Taylor series uncountable uniform convergence union