Peano's Axioms
see definitions
Lemma 1.1
If n E N and n isn't 1, then there exists m E N s.t. o-(m) = n
Lemma 1.2 (Associativity)
If x, y, z E N, then x + (y + z) = (x + y) + z
Lemma 1.3 (Commutativity of addition)
For any x, y E N, x + y = y + x
Lemma 1.4 (Distributive law)
for all x, y, z E N, x(y + z) = xy + xz
Lemma 1.5
If x, y, z E N and x < y and y < z then x < z
Lemma 1.6
Let x, y, z E N; 1. If x </= y and y < z, then x < z; 2. If x < y and y </= z, then x < z; 3. If x </= y and y </= z, then x </= z
Lemma 1.7
For any m E N, m doesn't equal m + 1
Lemma 1.8
For any m, k E N, m doesn't equal m + k
Lemma 1.9 (Well ordering of N)
If n, m E N, then exactly one of the following is true. Either n < m, n = m, m < n
Lemma 1.10
Let S ( N. If S has a least element then it is unique; " the least element", not "a least element
Theorem 1.1
If S ( N and S is not empty, then S has a least element
Lemma 1.11
Given an equivalence relation on a set S, for any a, b E S, either [a] ^ [b] = 0 or [a] = [b]
Lemma 1.12
Let ~ be an equivalence relation on a set A. Then there exists a function f: A -> P(A), the power set of A so that ~ is the equivalence relation associated to f
Lemma 2.1
The addition defined for integers is well-defined
Lemma 2.2
Let A = [(a,b)]. Then A is positive iff b < a
Lemma 2.3
A is positive iff A = f(k) for some natural number k
Lemma 2.4
A E Z is positive iff -A is negative
Lemma 2.5
If a, b, c E N with a < b, then ac< bc
Lemma 2.6
Let a, b E N and 1 < a. Then there exists a natural number N s.t. for all n >/= N, b < a ^n
Lemma 2.7
For any a E Z, we have -IaI </= a </= IaI. conversely, if c >/= 0 is an integer and -c</= a </= c, then IaI < c
Lemma 2.8 (Triangle Inequality)
If a, b E Z, then Ia+bI </= IaI + IbI
Lemma 3.1
If A is any rational number, then A = [(a, b)] for some a, b E Z with b positive
Lemma 4.1
Let {xn} be a CS. Then there exists a rational number M E Q s.t. IxnI < M for all n; (the sequence is bounded)
Lemma 4.2
If {xn} and {yn} are CS, then so are {xn + yn} and {xnyn}
Lemma 4.3
Addition of real numbers as defined is well defined
Lemma 4.4
If A < B are two real numbers, there is a rational number q s.t. A < q < B.; a rational number q as an element of R is just teh equivalence class of the Cauchy sequence with all terms equal to q
Theorem 4.1
Let S ( R be a non-empty subset and assume that it has a lower bound M. Then there exists a real number & (called the infimum of S) such that for any s E S, s >/= & and if x E R is s. t. s >/= x for all s E S, then & >/= x
Theorem 4.2
Let S ( R be a nonempty subset and assume it has an upper bound. Then there exists a real number { (called the supremum of S) such that for any s E S, s </= { and if x E R is such that s </= x for all s E S, then { </= x
Lemma 4.5
Let S be a non-empty set bounded below and let & be its infimum. Then { is a lower bound for S. Further, given any positive number e, there exists an s ES such that & </= s < & + e. Similarly if S is a non-empty set bounded above, then its supremum { is an upper bound for S. Further given any positive number e, there exists an s E S s. t. { - e < s </= {
Lemma 4,6
The two definitions given for a Cauchy sequence of real numbers are equivalent
Theorem 4.3
Let {xn} be a Cauchy sequence of real numbers. Then there exists a unique real number x s. t. given any e > 0, real number, there exiss an N E N s.t. for all n >/= N Ixn - xI < e
Theorem 4.4
Let S be an infinite bounded set. That is, S is infinite and there exists an M > 0 s. t. for all s E S IsI </= M. then there exists an x E R s.t. for any e > 0, there are infinitely many elements s E S s.t. Ix - sI < e
Lemma 5.1
If f, g are continuous functions on R, so is f + g and fg, where (f + g)(x) = f(x) + g(x) and (fg)(x) = f(x)g(x) for all real numbers x
Lemma 5.1
If f is continuous at x and {xn} is any CS with lim xn = x, then lim f (xn) = f(x)
Theorem 5.1
Let f be a function (from R or an open interval to R). Then f is continuous at a point x iff, given any e > 0, there exists a % > 0 s.t. for any t s.t. I t-x I < %, we have If(t) -f(x)I < e
Theorem 5.2 (Intermediate Value Theorem)
Let f be a continuous function (on R or in the closed interval [a,b]) where a < b. Then given any real number u between f(a) and f(b, there exists a real number a </= c </= b s.t. f(c) = u