6 (4k – 1) = n(2n + 1)k=1for all n > 1.2. .Let U = (0, 1) U (2, 3) U (4, 5) be a union of three openintervals and A= [0, 10] \ U. Show that A is compact.Use the definition to show that4n –...

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Extracted text: Use mathematical induction to show that >(4k – 1) = n(2n + 1) k=1 for all n > 1. 2. . Let U = (0, 1) U (2, 3) U (4, 5) be a union of three open intervals and A= [0, 10] \ U. Show that A is compact. Use the definition to show that 4n – 5 lim n+00 2n + 3 = 2. 4. Use the definition to show that x + x – 6 lim = 5. x - 2 5. Calculate the following limits lim n(vn2 +1-n), lim 0+ sin Show that the function f(x) = a is uniformly continuous n00 6. on [10, 20] but not uniformly continuous on [10, +o0). (2x, +7)/4 Define a sequence {xn} by x1=2 and xn+1= for n = 1, 2, 3, .. Show that {rn} is convergent and find its limit. 7. 8. Show that cos a- cos y < |æ="" –="" y|="" for="" all="" a="" and="" y.="">