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MAT2125 Summary.pdf

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Barry Jessup

MAT2125 Summary A number is rational, or in , if and only if it starts repeating itself eventually, or if the decimal expansion ends. Working definition of the real numbers: { } F ACTS ABOUT INEQUALITIES 1. then or 2. and implies that 3. 4. and means that 5. and means that 6. If and i. , then ii. then 7. If , then Absolute value of if is| | { , or| | | | | | | | Triangle Inequality: , "The absolute value of is less than or equal to the absolute value of plus the absolute value of . Archimedean Property: , such that . For all rational numbers, there exists a natural number that is smaller than or equal to the rational number. A set is countable if is infinite and there is a bijection such that 1. (injective) 2. such that (surjective) R EAL N UMBERS : 1. is a field whose operations agree with those on since . 2. is an ordered field, and the order agrees with that on . 3. The function | | | satisfies the same properties as the rational numbers. In particular, , | | | | | | 4. has the Archimedean property. such that 5. is a complete ordered field, i.e. every Cauchy sequence of real numbers converges to a real number Let . A real number is an upper bound for if , . If has an upper bound, we say it is bounded above. Let be bounded above. A real number is a least upper bound or a supremum of if for all upper bounds of , , where is an upper bound. If has a supremum, we write . The supremum is unique. Theorem: Let be bounded above. Then, if and only if: 1. , . 2. with Completeness Every nonempty subset which is bounded above has a supremum, which is a Theorem: real number. Lemma: Every interval with contains at least one integer. Density is dense in . This means that every open interval in with contains a Theorem: rational number. Corollary: Given any real number and any with | | , i.e. If (all things in that are not in ), we say that is irrational. is dense in , i.e. every open interval with contains an irrational number. Suppose a subset of reals is bounded below, i.e. such that . A real number is an infimum or a greatest lower bound for if 1. is a lower bound for 2. for all being lower bounds. Equivalently, if 1. 2. , with If is nonempty and is bounded below, then exists, and it is unique. S EQUENCES AND T HEIR L IMITS A sequence of real numbers is a function . We write and will henceforth say "Let be a sequence." The sequence converges to a real number if such that | | . If such an exists, we write or or as . Remarks: 1. When converges, the will eventually depend on the 2. The order of the expressions in the definition is crucial! If no such exists, we say that the sequence does not converge, or the sequence has no limit, or the limit doesn't exist. Theorem: If and , then . This says that if the limit exists, it is unique. Infinite Limits If is a real sequence, we say that 1. if such that . 2. if such that . Theorem 1.3: If , then is bounded, i.e. such that | | . Note that not every bounded sequence converges. Useful facts: Suppose and . Then  If , then  If , then  If , define Then, . So if , such that , . So define { then Squeeze Suppose for some , three sequences satisfy: Theorem: 1. 2. , Then, Cauchy Sequence: Suppose . Let . Then there exists an such that for all , | | , so indeed | | if both . Definition: such that | | . Theorem 1.8: A sequence converges if and only if it is a Cauchy sequence. Theorem 1.13: Every increasing sequence which is bounded above converges to its supremum, i.e. if and for some , then | . Corollary: Every decreasing sequence which is bounded below converges to its infimum. Corollary 1.14: Suppose and and satisfy 1. 2. Then, 1. If converges, so does 2. If does not converge, neither does Theorem: Every Cauchy sequence is bounded, above and below. L IMIT SUPERIOR AND L IMIT INFERIOR Let be a bounded sequence. Then | is an increasing sequence which is bounded above (because is bounded above). Hence, exists. Similarly, if we define | , then and is bounded below. Hence, such that . Furthermore, , , and so, If is a bounded sequence, the limit inferior of is and the limit superior of is . The sequence converges if and only if , when converges, S UBSEQUENCES AND A CCUMULATION P OINTS If is a sequence, a sequence is a subsequence of if there is a function such that 1. 2. We usually write , and . Theorem: Nested Intervals. Suppose a sequence of intervals[ | satisfies: a.[ ] [ ] [ ] b. Then and ⋂ [ ]. Theorem: Bolzano-Weierstrass. A bounded sequence has at least one accumulation point, i.e. every bounded sequence has at least one convergent subsequence. S ERIES If is a sequence, define another sequence called the partial sums by ∑ .  is called the th partial sum of the series . The series∑ converges if and only if converges.   If , we write∑ .  If does not converge, we say that the series diverges. Harmonic series: ∑ diverges, or∑ Note that ∑ converges if and only if is Cauchy if and only if , such that ,| | Fact:∑ converges implies that . Note that the converse is not true. Theorem: Alternating Series Test. Suppose , and . Then ∑ converges, and |∑ | . Theorem: Comparison Test. If such that then: 1. ∑ converges ∑ converges. 2. ∑ diverges ∑ diverges. Theorem: Ratio Test. Suppose and such that for : 1. If , then ∑ converges. 2. If , then∑ diverges. If either is 1, anything can happen. Theorem: Root Test. Suppose is eventually positive (which means such that , ). If √ , and: 1. , then∑ converges. 2. , then∑ diverges. 3. , then∑ converges. Series with Arbitrary Terms: A series converges absolutely if | | converges. Proposition: A series which is absolutely convergent, converges. Corollary: 1. If a series is absolutely convergent, then both the series formed by its positive terms and the series of its negative terms are convergent. The series converges to the sum of these two. 2. If a series converges but not absolutely, then neither series formed by the positive or negative terms can converge. In other words, they both diverge. Theorem: If a series is absolutely convergent, then the series formed by any rearrangement of its terms converges to the same sum. Rearrangement: If ∑ is a series, a rearrangement of is∑ where is any bijection. T OPOLOGY OF , SEQUENCES IN The Euclidean norm on is a function || || such that (among other properties) it satisfies ‖ ‖ √ . For all , : 1. ‖ ‖ if and only if 2. ‖ ‖ | |‖ ‖ 3. ‖ ‖ ‖ ‖ ‖ ‖ (Triangle Inequality) 4. ‖ ‖ ‖ ‖ ‖ ‖ (Cauchy-Schwartz Inequality) A couple of popular norms:  Box norm: | | | |  Diamond norm: ∑ | | Definitions: Fix . Let be a sequence of vectors in . If , then we say if | || and only if such that , (limit). Also, is Cauchy if such that | | . Remarks: Let 1. | | √ √ ‖ ‖ o Shows that if and , then for each , . In other words, . 2. ‖ ‖ ‖ ‖ ∑ ‖ ‖where is a vector with a 1 in the th place and 0 in the remaining places. In other words, . But‖ ‖ | |‖ ‖ | |,in other words‖ ‖ ∑ | |. o Shows that if , then . Moreover, is Cauchy if and only if each of its compoments is Cauchy. Theorem 1.8: Cauchy in . A sequence of vectors in converges if and only if the sequence is Cauchy. Theorem 1.9: Bolzano-Weierstrass for . Every bounded sequence in has a convergent subsequence. Note: Every subsequence of a subsequence is a subsequence of . O PEN SETS AND O PEN B ALLS Let . The open ball, center , of radius , is { | ‖ ‖ }.  When , it's called an open interval.  When , it's called an open disc.  When , it's called an open ball. Note that open balls are open sets. In other words, such that . A set in , call it , is an open set if such that . The just means that the might depend on . Examples:  Open intervals where .  and are open.  is not open, and is not open.  Any union of open sets is open.  A finite intersection of open sets is open. o Note that some infinite intersection of open sets are not open. o Example: ⋂ ( ) [ because 0 has no "elbow room." A set in , call it , is a closed set if | is open.  and are closed.  Closed interva[s ]are closed because the complement is open.  Finite unions of closed sets are closed. o Infinite unions of closed sets may not be closed. o Example: ⋃ ( ) [ . Not closed since [ , the complement, isn't open since 1 has no "elbow room".  Any intersection of closed sets is closed. Complementsof intersections are unions of complements. Theorem: Definition 1.12. A set is closed if and only if every sequencewhich converges has its limit in . Theorem: A subset is compact if every sequence in has an accumulation point in . is compact if and only if is closed and bounded.  Every closed finite interval [ ] is compact.  Every finite set in is comp
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