The Real And Complex Number Systems

More documents

Recommendations

Info

Remark: The reader may give it a try to show that lim n→ sup sin n 1 and lim n→ inf sin n −1. (b) 1 1 n cosn Proof: Note that So, it is clear that lim n→ sup 1 1 n 1 1 n cosn 1ifn 2k −1 ifn 2k − 1 . cosn 1 and lim n→ inf 1 1 n cosn −1. (c) n sin n 3 Proof: Note that as n 1 6k, n sin n 1 6k sin ,andasn 4 6k, 3 3 n −4 6k sin . So, it is clear that 3 lim n→ sup n sin n and lim 3 inf n sin n n→ 3 −. (d) sin n cos n 2 2 Proof: Note that sin n cos n 1 sin n 0, we have 2 2 2 lim n→ sup sin n 2 cos n 2 lim inf sin n n→ 2 cos n 2 0. (e) −1 n n/1 n n Proof: Note that we know that (f) n 3 − n 3 Proof: Note that So, it is clear that lim n→ −1 n n/1 n n 0, lim n→ sup−1 n n/1 n n lim n→ inf−1 n n/1 n n 0. n 3 − n 3 1 3 2 3 if n 3k 1 if n 3k 2 0ifn 3k ,wherek 0, 1, 2, . . . . lim n→ sup n 3 − n 3 2 and lim 3 inf n n→ 3 − n 3 Note.In(f),x denoted the largest integer ≤ x. n 8.8 Let a n 2 n − ∑ k1 1/ k . Prove that the sequence a n converges to a limit p in the interval 1 p 2. n n Proof: Consider ∑ k1 1/ k : S n and x −1/2 dx : T n , then 1 lim n→ d n exists, where d n S n − T n by Integral Test. We denote the limit by d, then 0.
0 ≤ d 1 * by Theorem 8.23 (i). Note that d n − fn is a positive increasing sequence, so we have d 0. ** Since T n 2 n − 2 which implies that lim n→ n 2 n − ∑ 1/ k lim n→ a n 2 − d p. k1 By (*) and (**), we have proved that 1 p 1. Remark: (1) The use of Integral Test is very useful since we can know the behavior of a given series by integral. However, in many cases, the integrand may be so complicated that it is not easy to calculate. For example: Prove that the convergence of ∑ n2 1 nlog n p ,wherep 1. Of course, it can be checked by Integral Test. But there is the Theorem called Cauchy Condensation Theorem much powerful than Integral Test in this sense. In addition, the reader can think it twice that in fact, Cauchy condensation Theorem is equivalent to Integral Test. (Cauchy Condensation Theorem)Let a n be a positive decreasing sequence. Then ∑ n1 a n converges if, and only if, ∑ k0 2 k a 2 k converges. Note: (1) The proof is not hard; the reader can see the book, Principles of Mathematical Analysis by Walter Rudin, pp 61-63. (2) There is an extension of Cauchy Condensation Theorem (Oskar Schlomilch): Suppose that a k be a positive and decreasing sequence and m k ⊆ N is a sequence. If there exists a c 0 such that 0 m k2 − m k1 ≤ cm k1 − m k for all k, then ∑ k1 a k converges if, and only if, ∑m k1 − m k a mk . k0 Note: The proof is similar with Cauchy Condensation Theorem, soweomitit. (2) There is a similar Theorem, we write it as a reference. If t ≥ a, ft is a non-negative increasing function, then as x ≥ a, wehave x ∑ fn − ftdt a≤n≤x a ≤ fx. Proof: The proof is easy by drawing a graph. So, we omit it. P.S.: The theorem is useful when we deal with some sums. For example, ft log t. Then
Page 1 and 2:
The Real And Complex Number Systems
Page 3 and 4:
Proof: Consider 2n−2 ∑ (x + 1)
Page 5 and 6:
Then S = N. Proof: (A ⇒ B): If S
Page 7 and 8:
Note: There are many and many metho
Page 9 and 10:
In addition, (√ a ) 2 − − b (
Page 11 and 12:
Proof: Say {ar + b : a ∈ Z, b ∈
Page 13 and 14:
(⇐)It is clear since every a n is
Page 15 and 16:
Proof: Choose a 0 = [x], and thus c
Page 17 and 18:
Proof: Use Cauchy-Schwarz inequalit
Page 19 and 20:
In addition, ∑ a k b k + a j b j
Page 21 and 22:
So, z 1 z 2 = ¯z 1¯z 2 (c) z¯z =
Page 23 and 24:
so, |a − b| 2 ≤ ( 1 + |a| 2) (
Page 25 and 26:
Proof: Note that in this text book,
Page 27 and 28:
1.39 State and prove a theorem anal
Page 29 and 30:
1.43 (a) Prove that Log (z w ) = wL
Page 31 and 32:
For complex θ, we show that it hol
Page 33 and 34:
So, ix−y = i (x + iy) = iz = log
Page 35 and 36:
and the sum of their squares is giv
Page 37 and 38:
Some Basic Notations Of Set Theory
Page 39 and 40:
(i) If x ∈ R, it is clear that (x
Page 41 and 42:
(a) A ∪ (B ∪ C) = (A ∪ B) ∪
Page 43 and 44:
(e) A ∩ (B − C) = (A ∩ B) −
Page 45 and 46:
Proof: Given x ∈ X, then f (x)
Page 47 and 48:
By hyppothesis, we get T ⊆ f (S)
Page 49 and 50:
Proof: Since S is an infinite set,
Page 51 and 52:
Proof: Assume S˜R and let f be a o
Page 53 and 54:
2.22 Let S denote the collection of
Page 55 and 56:
Charpter 3 Elements of Point set To
Page 57 and 58:
And thus by the remark in (e), it i
Page 59 and 60:
3.7 Prove that a nonempty, bounded
Page 61 and 62:
Bx, r x S Bx, r x T . So, at
Page 63 and 64:
Claim that Bp, r S as follows. Let
Page 65 and 66:
Covering theorems in R n 3.17 If S
Page 67 and 68:
uncountable. Hence, by exercise 3.2
Page 69 and 70:
Then we have (a) (b) (c) (d) x y
Page 71 and 72:
(b) Give an example of a metric spa
Page 73 and 74:
Hence from (1)-(4), we know that i
Page 75 and 76:
3.44 intM A M A . Proof: Let B
Page 77 and 78:
That is, intB which is absurb. He
Page 79 and 80:
Limits And Continuity Limits of seq
Page 81 and 82:
x n1 x n 1 1 x n x n 1 2 . Rema
Page 83 and 84:
|a m a n| |a m a m1 a m1 a m2
Page 85 and 86:
and use the fact if a n converges
Page 87 and 88:
lim y0 fx, y 0ifx 0, 1ifx 0. and
Page 89 and 90:
fx, y 1 r cos sinr2 cossin if x
Page 91 and 92:
then we have lim fx, y L if x 0an
Page 93 and 94:
this is because a can be an isolate
Page 95 and 96:
irrational. Prove that: (a) ffx x
Page 97 and 98:
In addition, since f1 f1 by f0 0,
Page 99 and 100:
lim r n0 fx f lim n x n lim n fx
Page 101 and 102:
if x Q, andgx 0ifx Q c . Remark:
Page 103 and 104:
number f T supfx fy : x, y T is
Page 105 and 106:
Proof: Since the hypothesis says th
Page 107 and 108:
(f) S 0, 1 0, 1, T 0, 1 0, 1. S
Page 109 and 110:
In Exercises 4.29 through 4.33, we
Page 111 and 112:
function. Since f0, 1 0, 1 0, 1,
Page 113 and 114:
4.40 If x is a point in a metric sp
Page 115 and 116:
from b to A) are disjoint. So, if e
Page 117 and 118:
x F k1 F k A B U V which im
Page 119 and 120:
Proof: Assume that B C 1 is discon
Page 121 and 122:
Proof: By Mean Value Theorem, wehav
Page 123 and 124:
x r y r rz r1 x y ry r1 /2. So,
Page 125 and 126:
hx gfx if x S. Iff is uniformly c
Page 127 and 128:
dx, y , x, y A, wehave dfx, fy
Page 129 and 130:
In addition, part (b) comes from in
Page 131 and 132:
continuous at a, a. (2) (x y) Sinc
Page 133 and 134:
Proof: Let D denote the set of dico
Page 135 and 136:
(a) Provet that the formula df, g
Page 137 and 138:
so, by induction, we find dp n1 , p
Page 139 and 140:
Proof: If p and p are fixed points
Page 141 and 142:
(b) Assume there is a subsequence p
Page 143 and 144:
Hence, f is increasing on , a/3 a
Page 145 and 146:
f k x f k 0 x 0 fk x x by induct
Page 147 and 148:
h k1 h k k jk f j g kj x j0
Page 149 and 150:
g x g x 3 2 a d f x a d f x 3 2
Page 151 and 152:
which implies that F 0, where
Page 153 and 154:
g fx gfx. Assume that there is a
Page 155 and 156:
Remark: For x 0, we can show that
Page 157 and 158:
Proof: Look at the Generalized Mean
Page 159 and 160: Proof: By Generalized Mean Value Th
Page 161 and 162: there is a point p such that fp 0.
Page 163 and 164: Remark: If we can make sure that fx
Page 165 and 166: which implies that Hence, g 1 h g
Page 167 and 168: fx fy x y f x /2 *’ Combi
Page 169 and 170: lim xa fx gx L. Remark: 1. The pr
Page 171 and 172: every g k is never zero in c , c
Page 173 and 174: f x 1 ...x n n f x 1 ...x n1 n x
Page 175 and 176: f x 1 ...x n n fx 1 ...fx n n . C
Page 177 and 178: lx fc f cx c fx. (Exercise 2)
Page 179 and 180: So, we have Partial derivatives fb
Page 181 and 182: ux, y ey e y cosx 2 and vx, y e
Page 183 and 184: ux, y (1) arctany/x, ifx 0, y R
Page 185 and 186: Functions of Bounded Variation and
Page 187 and 188: For the part fx x fy 1 |fx fy| 1
Page 189 and 190: Claim that 1 sup A. Suppose NOT, t
Page 191 and 192: we know that V f is an increasing
Page 193 and 194: in Section 6.12. Proof: Sinceft : t
Page 195 and 196: 2n h P 2 hx i hx i1 i1 n 2n
Page 197 and 198: n k1 n ||fb k | |fa k || |fb k
Page 199 and 200: Supplement on lim sup and lim inf I
Page 201 and 202: (1) lim n→ inf a n − a n has
Page 203 and 204: Remark: The exercise is useful in t
Page 205 and 206: which implies that c ≤ a b since
Page 207 and 208: If lim sup n→ a n1 a n , then it
Page 209: We first prove lim n→ sup n ≤
Page 213 and 214: Consider which implies that which i
Page 215 and 216: So, L 1 5 since L ≥ 1. 2 Remark:
Page 217 and 218: For case (i), since 1 n log nlog lo
Page 219 and 220: ()Assume that ∑ n1 That is, p!
Page 221 and 222: n S n ∑−1 k1 1 3k − 2 − 1
Page 223 and 224: (b) Prove that ∑ n1 −1 n−1 /
Page 225 and 226: then b k sin kx, a k1 − a k 1
Page 227 and 228: diverges, then ∑ where By (2) and
Page 229 and 230: Suppose NOT, it means that lim n→
Page 231 and 232: then since n lim n→ ∑ k1 n S n
Page 233 and 234: |sin k|r ∑ k So, ∑ |sink|r dive
Page 235 and 236: n For the part 1 2 0 that x sint
Page 237 and 238: lim n→ sin ... sin n 1 ... 1 n
Page 239 and 240: Also, lim q→ fp, q lim q→ p si
Page 241 and 242: n c n ∑ a k b n−k k0 n ∑ k0
Page 243 and 244: So, n s n ∑ cos2k − 1x j1 sin
Page 245 and 246: n u n ∑−1 k a k k1 n ∑−1
Page 247 and 248: which implies that which implies th
Page 249 and 250: So, by above sayings, we have prove
Page 251 and 252: (a) If fn is multiplicative and if
Page 253 and 254: Sequences of Functions Uniform conv
Page 255 and 256: (b) Prove that h n (x) does not con
Page 257 and 258: Proof: Since g is continuous on a c
Page 259 and 260: Hence, {g (x) x n } converges unifo
Page 261 and 262:
y continuity of f k(x0 ) (x) − f
Page 263 and 264:
In addition, as c ≥ 1/2, if f n
Page 265 and 266:
(Lemma) If {a n } and {b n } are tw
Page 267 and 268:
9.16 Let {f n } be a sequence of re
Page 269 and 270:
which implies that ∣ lim sup 1 k
Page 271 and 272:
have ∫ x 0 ∞∑ t 2n − 1 2 n=
Page 273 and 274:
[ ] π And as x ∈ , π , then n+p
Page 275 and 276:
0.1 Supplement on some results on W
Page 277 and 278:
which implies that h (x + t) − h
Page 279 and 280:
we have ∫ d c |f n − g| 2 dx
Page 281 and 282:
we complete it. 9.31 Given that two
Page 283 and 284:
if x = λ (≠ 0) is a root of 1 +
Page 285 and 286:
So, the series diverges. (ii) As
Page 287 and 288:
9.38 For each real t, define f t (x
Page 289 and 290:
So, B 0 = P 0 (0) = C 0 = 1, B 1 =
Page 291 and 292:
Remark: (1) The reader can see the
Page 293 and 294:
That is, lim n inf a n sup c n . n
Page 295 and 296:
and lim n infa n b n lim n a n
Page 297 and 298:
q! kq1 1 k! kq1 q! k! 1 q 1
Page 299:
g x log 1 1 x x a x 2 x log
show all

The Real And Complex Number Systems

Create successful ePaper yourself

Delete template?

Save as template?