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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 contains points continuous function converges uniformly convex countable number countable set covered decimal defined definition dense set diam diameter differentiable digits Dini derivates discontinuous disjoint distance elements empty example Exercise exists f is continuous finite number function f given graph Heine-Borel theorem Hence horizontal chord inequality inﬁnite interior point intermediate value property intersection inverse image least upper bound Let f lim sup limit point lower bound Math 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 rational numbers rational points real numbers right-hand set of measure set of points Similarly subinterval subset Taylor series uncountable uniform convergence union