課程名稱：高等微積分一
課程性質︰數學系必修
課程教師︰陳俊全
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2010/11/23
考試時限（分鐘）：10:20 ~ 13:20
是否需發放獎勵金：是
Choose 5 from the following 8 problems.
(取前五高分答題，每題20分)
編按：(i) 為了避免與箭頭混淆，小於等於我寫成 =<
(ii) 序列(sequence) Xn 的寫法我改成用底線分隔: x_n
(iii) 佔據兩格的Ｑ與Ｒ分別代表有理數與實數
1. Use the axioms of an ordered field (please see below) to prove the
following properties:
(a) 0*x = 0 for every x.
(b) (-1)*x = -x.
(c) If x*y = 0, then x = 0 or y = 0.
(d) If x ≠ 0, then x^(-1) ≠ 0 and (x^(-1))^(-1) = x.
(e) x^2 >= 0 for any x.
┌────────────────────────────────────┐
│ Axioms of an ordered field: │
│(1) x+y = y+x. (2) x+(y+z) = (x+y)+z. (3) ∃0 such that x+0 = x. │
│(4) For each x, ∃-x such that x+(-x) = 0. │
│(5) x*y = y*x. (6) x*(y*z)=(x*y)*z. (7) ∃1 such that 1*x = x. │
│(8) For each x ≠ 0, ∃x^(-1) such that x*(x^(-1)) = 1. │
│(9) x*(y+z) = x*y + x*z. (10) 1 ≠ 0. │
│(11) x=<x. (12) x=<y, y=<x => x=y. (13) x=<y, y=<z => x=<z. │
│(14) For each pair x,y, either x=<y or y=<x. (15) x=<y => x+z =< y+z. │
│(16) 0=<x, 0=<y => 0=<x*y. │
└────────────────────────────────────┘
2. A subset E of Ｑ is called a Dedekind cut if the following properties
hold: (1) E≠ø, E≠Ｑ; (2) if p∈E, q∈Ｑ, and q<p, then q∈E;
(3) If p∈E, then p<r for some r∈E. Let R' be the collection of all
of the Dedekind cuts.
(a) Let E and F be two cuts. Define E =< F to mean E⊆F. Show that R'
satisfies the least-upper-bound property (LUBP).
(b) For E,F∈R', define E+F to be the set {r+s|r∈E, s∈F}. Show that
E+F∈R'.
3. Let x_n and y_n be two bounded sequences in Ｒ.
(a) Prove that limsup(x_n) = inf{sup{x_(n+1),x_(n+2),...}|n=1,2,...}.
(b) Prove that limsup(x_n+y_n) =< limsup(x_n) + limsup(y_n). Show
that the equality holds if x_n is convergent.
4. Prove that a sequence x_n in Ｒ converges if and only if it is a
Cauchy sequence.
5. For a set E in a metric space M and x∈M, let
d(x,E) = inf{d(x,y)|y∈E},
and for ε> 0, let D(E,ε) = {x|d(x,E)<ε}.
(a) Show that D(E,ε) is open.
(b) Let E⊂M and Nε= {x|d(x,E)=<ε}, where ε>0. Show that Nεis
closed and that E is closed if and only if E = ∩{Nε|ε>0}.
6. (a) Let E be a bounded subset of Ｒ^2. Show that bd(E) is
uncountable if E contains an interior point.
(b) Find an open set E in Ｒ^2 such that bd(E) is finite.
(c) Find a countable set E in Ｒ^2 such that bd(E) is uncountable.
7. Use that ε-Ｎ definition to prove the following statements.
(a) Let lim x_n = A∈Ｒ. Show that
n->∞
lim (x_1 + x_2 +...+ x_n)/n = A.
n->∞
(b) lim n^(1/n) = 1.
n->∞
8. Let E = {(x,y)∈Ｒ^2 | y > x^2}. Use the definition of an open set
and the definition of the boundary of a set to show that E is open
in Ｒ^2 and bd(E) = {(x,y) | y = x^2}.
----
已經校正過一次
但若有任何錯誤或看不見的字
還煩請站內信告知
感謝
(別寄信問詳解阿= =
我才寫三題半)
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 118.167.186.102