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Rudin's Principles of Mathematical Analysis: Solutions to ... - MIT

Rudin’s **Principles** **of** **Mathematical** **Analysis**: **Solutions** **to** Selected Exercises Sam Blinstein UCLA Department **of** Mathematics March 29, 2008

- Page 2 and 3: Contents Chapter 1: The Real and Co
- Page 4 and 5: Solution: To show part (a) we simpl
- Page 6 and 7: ordered set and in this case z < u.
- Page 8 and 9: which immediately implies that ac +
- Page 10 and 11: 17. Prove that |x + y| 2 + |x − y
- Page 12 and 13: 3. Prove that there exist real numb
- Page 14 and 15: (f) Do E and E ◦ always have the
- Page 16 and 17: 1/n ∈ B(0, δ) ⊂ Uα0 . Thus, w
- Page 18 and 19: Solution: s1 = √ 2 < 2. Suppose i
- Page 20 and 21: actually using an equivalent defini
- Page 22 and 23: 9. Omitted. 10. Omitted. 11. Suppos
- Page 24 and 25: thus �∞ an n=1 converges by The
- Page 26 and 27: 5. If f is a real continuous functi
- Page 28 and 29: Reversing the roles of xn and x we
- Page 30 and 31: since f achieves its maximum on the
- Page 32 and 33: (b) Prove that ρE(x) is a uniforml
- Page 34 and 35: 2. Suppose that f ′ (x) > 0 on (a
- Page 36 and 37: 7. Suppose f ′ (x), g ′ (x) exi
- Page 38 and 39: 11. Suppose f is defined in a neigh
- Page 40 and 41: y the definition of F (x, y, λ), h
- Page 42 and 43: 3. Omitted. 4. If f(x) = 0 for all
- Page 44 and 45: Therefore, taking integrals preserv
- Page 46 and 47: If I �= 0 and J �= 0 then let c
- Page 48 and 49: We can see the same result by using
- Page 50 and 51: and the series on the right can be
- Page 52 and 53:
Solution: Let ε > 0 be given. Sinc

- Page 54:
|fn(pi) − f(pi)| small, which can