notes1 - CM221A – 2010 ANALYSIS I NOTES ON WEEK 1...

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Unformatted text preview: CM221A – 2010 ANALYSIS I NOTES ON WEEK 1 NOTATION (not examinable) We shall use the following standard notation N is the set of positive integer numbers, N = { 1 , 2 ,... } . Z is the set of integer numbers, Z = { ...,- 2 ,- 1 , , 1 , 2 ,... } . Q is the set of rational numbers, Q = { m n : m,n ∈ Z } . R is the set of real numbers. ∞ is a shorthand for “infinity”. It is not a proper number. ∀ means “for all” or “for every”, ∃ means “there exists” or “there is”, The colon : in a mathematical formula means “such that”. REAL NUMBERS: AXIOMS (not examinable) Real numbers obeys the following axioms. ( A1 ) a + ( b + c ) = ( a + b ) + c for all a,b,c ∈ R ; ( A2 ) a + b = b + a for all a,b ∈ R ; ( A3 ) there is a unique element in R , denoted 0, such that a + 0 = a for all a ∈ R ; ( A4 ) for every a ∈ R , there is a unique element in R , denoted- a , such that a + (- a ) = 0; ( A5 ) ( a × b ) × c = a × ( b × c ) for all a,b,c ∈ R ; ( A6 ) a × b = b × a for all a,b ∈ R ; ( A7 ) there is a unique element in R , denoted 1, such that a × 1 = a for all a ∈ R ; ( A8 ) for every nonzero a ∈ R , there is a unique element in R , denoted a- 1 or 1 a , such that a × a- 1 = 1; ( A9 ) a × ( b + c ) = a × b + a × c for all a,b,c ∈ R . Definition. Substraction is defined by a- b = a + (- b ). Definition. Division is defined by a b = a × ( b- 1 ). Remark.- 1 does not exist. The expression a has no meaning. One “orders” two numbers by thinking of the larger as being the higher in or- der. Formally speaking, there is a relation < between elements of R obeying the following axioms. 1 ( A10 ) for any x,y ∈ R , exactly one of the following is true: either x = y , or x < y , or y < x ; ( A11 ) if x < y and y < c then x < c ; ( A12 ) if x < y then x + c < y + c for all c ∈ R ; ( A13 ) if x < y and c > 0 then xc < y c . Definition. We write x > y if y < x , x 6 y if either x < y or x = y (or, in other words, if it is false that x > y ) and x > y if y 6 x . Remark. ( A1 )–( A13 ) are axioms and cannot be proved. All other known equalities and inequalities involving the addition and composition can be deduced from the above axioms. Notation. For any x ∈ R , the modules (or absolute value) of x is defined by | x | = ( x, if x > 0,- x, if x < 0. We have | a | = |- a | , | ab | = | a || b | and | a- b | 6 | a | + | b | . If r > 0 then the inequality | x | < r is equivalent to the pair of inequalities- r < x and x < r . These results are proved by considering all possible cases of positive and negative a , b and a + b and applying the axioms ( A1 )–( A13 ) (see the online lecture notes for details)....
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notes1 - CM221A – 2010 ANALYSIS I NOTES ON WEEK 1...

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