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### 48 Cards in this Set

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 Well-Ordering Principle of N Every nonempty subset of N has a least element. Algebraic properties of R (A1)a+b=b+a (A2)(a+b)+c=a+(b+c) (A3)0+a=a (A4)a+(-a)=0 (M1)a*b=b*a (M2)(a*b)*c=a*(b*c) (M3)1*a=a (M4)a*(1/a)=1 (D)a*(b+c)=(a*b)+(a*c) The order properties of R There is a nonempty subset P of R, called the positive real numbers, with these properties: (1)If a,b belong to P, then A+B belongs to P. (2)If a,b belong to P, then ab belongs to P. (3)Trichotomy Property Trichotomy Property If a belongs to R, then one of the following holds: a∈P, a=0, -a∈P The notion of inequality btw 2 real numbers Let a,b be elements of R. (a)If a-b∈P, then a>b or bb and b>c then a>c 2)If a>b then a+b>b+c 3)If a>b and c>0 then ca>cb. 4)If a>b and c<0 then ca0 . 0 is a=0 . -a if a<0 Triangle Inequality If a,b∈R, then ∣a+b∣≤∣a∣+∣b∣ Two variations of the triangle inequality . ∣∣a∣-∣b∣∣≤∣a-b∣ . ∣a-b∣≤∣a∣+∣b∣ Additional(4) Properties of absolute value (a) ∣ab∣=∣a∣∣b∣ ∀ a,b∈R (b) ∣a∣^2 = a^2 ∀ a∈R (c) If c≥0, then ∣a∣≤c iff -c≤a≤c (d) -∣a∣≤a≤∣a∣ Let a∈R and ε>0. Then the ε-neighborhood of a is the set Vε(a):={x∈R:∣x-a∣<ε} For a∈R, the statement that x belongs to Vε(a) is equivalent to -ε0, then x=a Let S be a nonempty subset of R. The set S is said to be bounded above if ∃ a number u∈R s.t. s≤u ∀s∈S. u is called an upper bound of S. Let S be a nonempty subset of R. The set S is said to be bounded below if ∃ a number w∈R s.t. w≤s ∀s∈S. w is called a lower bound of S. Let S be a nonempty subset of R. A set is said to be bounded if it is both bounded above and bounded below. Let S be a nonempty subset of R. A set is said to be unbounded if it is not bounded. Let S be a nonempty subset of R. If S is bounded above, then a number u is said to be a supremum (a least upper bound) of S if it (1)u is an upper bound of S, and (2)if v is any upper bound of S, then u≤v sup S Let S be a nonempty subset of R. If S is bounded below, then a number w is said to be an infimum (a greatest lower bound) of S if it (1)w is a lower bound of S, and (2)if t is any lower bound of S, then t≤w inf S Archimedean Property IF x∈R, then ∃ nₓ∈N s.t. xb and b>c then a>c If a-b∈P and b-c∈P then (a-b)+(b-c)=(a-c)∈P therefore a>c Prove If a>b then a+b>b+c If a-b∈P then (a+c)-(b+c)=(a-b)∈P Prove If a>b and c>0 then ca>cb. If a-b∈P and c∈P then ca-cb=c(a-b)∈P therefore ca>cb when c>0 Prove If a>b and c<0 then caca when c<0 Completeness Property of R Every nonempty set of real numbers that has an upper bound also has a supremum in R. Limit of a sequence A sequence X=x(n) in R is said to converge to x∈R, or x is a limit of x(n), if for every ε>0 ∃ a natural number K(ε) s.t. for n≥K(ε), the terms x(n) satisfy ∣x(n)-x∣<ε Uniqueness of limits A sequence in R can have at most 1 limit Prove Uniqueness of limits Suppose lim(x(n))=X' and X". For each ε>0 ∃ K' s.t. ∣x(n)-X'∣<ε/2 for all n≥K' and ∃ K" s.t. ∣x(n)-X"∣<ε/2 for all n≥K". Let K be the larger of K' and K". Then for n≥K, with the Triangle Inequality, ∣X'-X"∣=∣X'-x(n)+x(n)-X"∣ ≤∣X'-x(n)∣+∣x(n)-X"∣ <ε/2+ε/2=ε Since ε>0 then X'-X"=0 Limit Theorems: X+Y Let X=x(n) and Y=y(n) be sequences of real numbers that converge to x and y sequence X+Y converges to x+y Limit Theorems: X-Y Let X=x(n) and Y=y(n) be sequences of real numbers that converge to x and y sequence X-Y converges to x-y Limit Theorems: X*Y Let X=x(n) and Y=y(n) be sequences of real numbers that converge to x and y sequence X*Y converges to xy Limit Theorems: cX Let X=x(n) be sequence of real numbers that converge to x and let c∈R sequence cX converges to cx Squeeze Theorem Suppose X=x(n), Y=y(n) and Z=z(n) are sequences of real numbers s.t. x(n)≤y(n)≤z(n) for all n∈N and the lim(x(n))=lim(z(n)). Then Y=y(n) is convergent and lim(x(n))=lim(y(n))=lim(z(n)) Monotone Sequence X=x(n) is a sequence then X is monotone if it either increases or decreases Bolzano-Weierstrass Theorem If an infinite set S is contained in a finite interval [a,b] then the set S has at least 1 limit point A in the interval [a,b] Limit Point A number x s.t. for all ε>0, ∃ a member of set y, different from x, s.t. ∣y-x∣<ε