[考題] 日/數學系/何清人/高等微積分
Demonhell的是95年的，這份是94年上學期期中考
1. State all the basic postulates of the real number system .
2. Prove that for all a,b belong to R with a<b and there exists r
belong to R such that a<r<b .
3. (i) State the Cauchy theorem .
(ii)Let {Xn} be a real sequence such that
1
| Xn+1 - Xn | ≦ ────
n(n+1)
for all n belong to N .Prove that {Xn} converges.
( PS.{Xn} ← n belong to N )
4. (i) Define limits supremum and infimum of a real sequence.
(ii) Suppose that Xn≧0 and Yn≧0. for all n belong to N.
Prove that :
if lim Xn exists , then lim sup(XnYn) =(lim Xn)(lim sup Yn) ,
n→∞ n→∞ n→∞ n→∞
provideed that none of these products is of the form 0．∞ .
5. (i) State the extreme value theorem .
(ii) If f：R→R is continuous and lim f(x) = lim f(x) = ∞ ,
x→∞ n→-∞
Prove that f has a minimum on R .
6. State and prove the intermediate value theorem .(用高微證法,勿用微積分版)
7. Let E ≦ R , and f：E → R , prove that f is uniformly continuous
(E包含於R)
on E <=> for any {Xn} and {Yn} in E with lim | Xn - Yn | = 0 ,
n→∞
we have lim |f(Xn) - f(Yn)| = 0 .
n→∞
( PS. {Xn}.{Yn} ← n belong to N )
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※ 編輯: shihweng 來自: 61.228.151.237 (05/09 02:07)
※ 編輯: shihweng 來自: 175.180.242.157 (07/04 19:11)
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