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) --

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