[미적분] 칼큘러스(미적분학) 증명에 대한 기본 예제들

소개글

칼큘러스 증명에 대한 기본 예제들에 대한 영문 자료.

목차

● Field axiom Exercise
● Oder axiom Exercise.
● Least-upper-bound axiom Exercise.
● Step function Exercise.
● Integral of step function.
● Integral of bounded function.
● Continuous function.

본문내용

● Field axiom Exercise
1.3.3.2. -0 = 0
1.3.3.5. -(a+b) = -a-b
1.3.3.7. (a-b) + (b-c) = a-c

● Oder axiom Exercise.
1.3.5.2. There is no real number x such that x2 + 1 = 0
1.3.5.4. If a>0, then 1/a > 0 ; if a<0, then 1/a < 0
1.3.5.10. If x has the property that 0≤ x< h for every positive real number h, then x=0.

● Least-upper-bound axiom Exercise.
1.3.12.1. If x and y are arbitrary real numbers with x<y, prove that there is at least one real z satisfying x < z < y.
1.3.12.3. If x > 0, prove that there is a positive integer n such that 1/n < x.
1.3.12.4. If x is an arbitrary real number, prove that there is exactly one integer n which satisfies the inequalities n ≤ x < n+1. This n is called the greatest integer in x and is denoted by [x].
1.4.10.16. The numbers 1, 2, 3, 5, 8, 13, 21, ...., in which each term after the second is the sum of its two predecessors, are called Fibonacci numbers. They may be defined by induction as follows:
a1=1, a2=2, an+1 = an+an-1 if n≥2
Prove that an < n for every n≥1.

참고 자료

Calculus