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Prove the real numbers are not countable. [Hint: they are. an ai2 an. a2i a22 α 2 3·. a3i a32 a 3 3.. /th digit in the decimal expansion of the ¿th number (arranged for counting) between 0 and 1. än ä22 (X33. . ] 7. Define a Cauchy sequence in a metric space. ) 8. Show directly from the definition that {l/n} is a Cauchy sequence. 9. Let , Xl = 1 , 1 X2 = - , 2' 3 Xz = - , * Z 4 4' "* = 5 8> · · · > - » Xn_2 + Xn-l 2 Show directly from the definition that {xn} is a Cauchy sequence. 48 III. Completeness Properties 10.

To verify this directly, consider I I / » - / I l = sup|/ n (aO -f(x) 0 0 for all n, and lim n ^ 00 / n (x) = /(x) = 0. Then, II/n — / | | = SUP |/»(x) - 0 | = 2y/n 50 III. Completeness Properties and lining || fn — / || = 0, so fn —> 0 in the norm of C[0, 1], as well as pointwise.

2. 10. 3. 11. 4. 12. 5. 9 to show that 6. 7. 12 in terms of {1/nJ 6 lv. (a) Show geometrically that ···. hChChC 1 / ■ * + 1l 1 dx X k+1 and hence 1 1 1 k~ " k + 1 ^ for every k = i, 2, . . , n, (b) Set 1 - - fk+l dx / — = c* and show l i m ^ « , ] ^ ^ ck exists (finite). 4. >o, 26 II. Sequence Spaces and Infinite (c) Find Ci + c2 + c3 + · · · + cn-i = Σ / L i c¿ (d) Show m Series closed form. lim I ( 1 + \ + I + - - - + - ) - In n 1 = lim Σ ck. n->ooL\ 2 3 nj J n^ k=1 This number is called Euler's constant or Mascheroni's constant.

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