The Real And Complex Number Systems

More documents

Recommendations

Info

fx y fx fy 1 x y 1 x 1 y 1 Q if x y Q c , x y 1 x 1 y 2 Q if x y Q. Remark: Here is an interesting question about functions. Let f : R 0, 1 R. Iff satisfies that fx f x 1 x 1 x, then fx x3 x 2 1 . 2xx1 Proof: Let x x1 x , then we have 2 x 1 x1 ,and3 x x. So, fx f x 1 x fx fx 1 x * which implies that fx f 2 x 1 x ** and f 2 x f 3 x f 2 x fx 1 2 x. *** So, by (*), (**), and (***), we finally have fx 1 2 1 x x 2 x x3 x 2 1 2xx 1 . 4.18 Let f be defined on R and assume that there exists at least one x 0 in R at which f is continuous. Suppose also that, for every x and y in R, f satisfies the equation fx y fx fy. Prove that there exists a constant a such that fx ax for all x. Proof: Let f be defined on R and assume that there exists at least one x 0 in R at which f is continuous. Suppose also that, for every x and y in R, f satisfies the equation fx y fx fy. Prove that there exists a constant a such that fx ax for all x. Proof: Suppose that f is continuous at x 0 , and given any r R. Since fx y fx fy for all x, then fx fy x 0 fr, wherey x r x 0 . Note that y x 0 x r, then lim xr fx lim xr fy x 0 fr yx0 lim fy x 0 fr fr since f is continuous at x 0 . So, f is continuous at r. Sincer is arbitrary, we have f is continuous on R. Definef1 a, andthensincefx y fx fy, wehave f1 f 1 m .. m 1 mtimes mf 1 m f 1 m f1 m *
In addition, since f1 f1 by f0 0, we have f1 f 1 m .. m 1 mf 1 m f 1 m mtimes f1 m *’ Thus we have f m n f1/m ...1/m ntimes nf1/m m n f1 by (*) and (*’) ** So, given any x R, and thus choose a sequence x n Q with x n x. Then fx f lim n x n lim n fx n by continuity of f on R lim n x n f1 by (**) xf1 ax. Remark: There is a similar statement. Suppose that fx y fxfy for all real x and y. (1) If f is differentiable and non-zero, prove that fx e cx ,wherec is a constant. Proof: Note that f0 1sincefx y fxfy and f is non-zero. Since f is differentiable, we define f 0 c. Consider fx h fx fh f0 fx fxf h h 0 cfx as h 0, we have for every x R, f x cfx. Hence, fx Ae cx . Since f0 1, we have A 1. Hence, fx e cx ,wherec is a constant. Note: (i) If for every x R, f x cfx, then fx Ae cx . Proof: Since f x cfx for every x, we have for every x, f x cfxe cx 0 e cx fx 0. We note that by Elementary Calculus, e cx fx is a constant function A. So,fx Ae cx for all real x. (ii) Suppose that fx y fxfy for all real x and y. Iffx 0 0forsomex 0 , then fx 0 for all x. Proof: Suppose NOT, then fa 0forsomea. However, 0 fx 0 fx 0 a a fx 0 afa 0. Hence, fx 0 for all x. (iii) Suppose that fx y fxfy for all real x and y. Iff is differentiable at x 0 for some x 0 , then f is differentiable for all x. And thus, fx C R. Proof: Since
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: irrational. Prove that: (a) ffx x
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 and 210:
We first prove lim n→ sup n ≤
Page 211 and 212:
0 ≤ d 1 * by Theorem 8.23 (i). 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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?