Bolzano Weierstrass Theorem For Sets

[Elly Garnand]

MAGY 6213 Intro to Math Analysis I3 Credits This course and its sequel MA-GY 6223 rigorously treat the basic concepts and results in real analysis. Course topics include limits of sequences, topological concepts of sets for real numbers, properties of continuous functions and differentiable functions. Important concepts and theorems include supremum and infimum, Bolzano-Weierstrass theorem, Cauchy sequences, open sets, closed sets, compact sets, topological characterization of continuity, intermediate value theorem, uniform continuity, mean value theorems and inverse function theorem. Offered in the fall.

Prerequisite(s): MA-UY 2122 or permission of adviser.

Weekly Lecture Hours: 3 Weekly Lab Hours: 0 Weekly Recitation Hours: 0

The Bolzano-Weierstrass Theorem is a fundamental theorem in real analysis, named after mathematicians Bernard Bolzano and Karl Weierstrass. It states that if a bounded infinite set of real numbers exists, then there exists at least one point in the set that has a convergent subsequence.

The Bolzano-Weierstrass Theorem is important because it provides a powerful tool for proving the convergence of sequences in real analysis. It also has applications in areas such as optimization, functional analysis, and differential equations.

The Bolzano-Weierstrass Theorem can be proved using various methods, such as the nested interval theorem, the bisection method, or the completeness axiom. However, the most common proof is known as the "diagonal argument" which involves dividing the set into two subsets and then recursively choosing subsequences from each subset until a convergent subsequence is found.

Yes, the Bolzano-Weierstrass Theorem can be extended to higher dimensions. In fact, the theorem is often stated in a more general form known as the Bolzano-Weierstrass Theorem for Euclidean spaces, which states that if a bounded infinite set exists in an n-dimensional Euclidean space, then there exists at least one point in the set that has a convergent subsequence.

Yes, there are some limitations to the Bolzano-Weierstrass Theorem. It only applies to bounded sets in real or Euclidean spaces, and it does not guarantee the uniqueness of the convergent subsequence. Additionally, the theorem cannot be applied to unbounded sets or sets in other types of spaces such as complex or metric spaces.

Numerical sets. The set of natural numbers. Induction principle. Bernoulli's inequality. Relative integers. Rational numbers. Existence of irrational numbers. The set of real numbers: algebraic structure, sorting. absolute value, powers and roots of a real number. Logarithms. The set of rational numbers is dense in the set of real numbers. Bounded sets of real numbers. Extrema of a numerical set and its properties. The extended straight. Intervals. The set of complex numbers. algebraic form, trigonometric form, powers and roots of a complex number.

Elements of topology in R. Neighbourhoods of a point. Interior, exterior, and boundary points. Interior and boundary of a set. Accumulation points. Derivative of a set. open sets, closed sets. Theorem of Bolzano-Weierstrass.

Real functions of one real variable. Definitions. Geometric representation. Extrema of a function. Definition of limit. Some examples. The unique limit theorem. Theorems of the comparison. Theorem of sign permanence. Left and right limits. Operations on the limits and indeterminate forms. Monotone functions and their limits. Infinitesimal and infinite. Asymptotes vertical, oblique or horizontal. Sequences and their limits. Characterization of the limit of a function by means of the limits of appropriate sequences. Monotone sequences. The number of Neper. Some known limits. Subsequences. Cauchy's convergence test. Sequentially compact sets and their characterization. Averages of the terms of a sequence.

Continuous functions. Definition of continuity in a point and in a set. Discontinuities. Discontinuities of monotone functions. Operations on continuous functions. Fundamental properties of continuous functions: theorem of the existence of zeros and of the existence of intermediate values, Weierstrass theorem. Continuity of composite functions and inverse functions. The functions arcsin x, arccos, arctan x. Uniform continuity. Cantor's theorem. Lipschitz continuity.

Differential calculus for real functions of one real variable. Derivative and its kinematic and geometric meanings. Differentiability and continuity. Derivatives of elementary functions. Rules of derivation. Derivatives of composite functions and inverse functions. Higher order derivatives. Maxima and minima, Fermat's theorem. The theorems of Rolle, Cauchy, and Lagrange. Some consequences of the Lagrange theorem: functions having zero derivative; characterization of the monotonicity for differentiable functions in an interval; functions with bounded derivative. Search of the maximum and minimum points of a function. Theorems of de l'Hospital and indeterminate forms. Convex functions in an interval. Property. Taylor's formula and applications. Study of a function's graph. Continuity of the derivative function.

Numerical series. Definitions and first properties. Cauchy convergence criterion. Geometric, Mengoli's, and harmonic series. Series with non-negative terms; convergence and divergence criteria: comparison, root, Raabe, and condensation criteria. Generalized harmonic series, criteria of infinitesimals. Absolute convergence. Series with terms having alternating sign, criterion of Leibniz. Operations on the series: sum, product by a constant. commutative property.

Integrals of real functions of one real variable. Integrability and integral of Riemann for limited functions in a closed bounded interval. A characteristic condition for the integrability and geometrical meaning. Example of function not Riemann integrable. Classes of integrable functions: continuous functions, monotonic functions, functions discontinuous at a finite set. Properties: distributivity, positivity, additivity, integration of the absolute value. The theorems of average. Definite integrals. Integral function and the fundamental theorem of calculus. Primitive and indefinite integrals. Elementary methods of indefinite integration: sum decomposition, by parts, by substitution. Integrals of rational functions. Integration for rationalization. Calculation of areas and volumes. Generalized integrals and improper integrals. Absolute integration, convergence criteria.

Notes on differential equations of the 1st and 2nd order. Model of Malthus. Damped harmonic oscillator. Differential equations of the 1st order with separable variables, homogeneous, linear and of Bernoulli's type. 2nd order differential equations, linear and with constant coefficients: structure of the space of solutions, method of variation of arbitrary constants.

A general remark that becomes clear quickly when one thinks about this is that there are fairly standard methods for converting one proof into another, and when we apply such a method then we tend to regard the two proofs as not interestingly different. For example, it is often possible to convert a standard inductive proof into a proof by contradiction that starts with the assumption that is a minimal counterexample. In fact, to set the ball rolling, let me give an example of this kind: the proof that every number can be factorized into primes.

The usual approach is the minimal-counterexample one: if there is a positive integer that cannot be factorized, let be a minimal one; is not prime, so it can be written as with both and greater than 1; by minimality and can be factorized; easy contradiction.

Now, equally briefly, here is another proof. Suppose again that and let be written in its lowest terms. Now . Substituting for and tidying up, this gives us that . But , so the denominator of the right-hand side is less than , which contradicts the minimality of (and hence too, since their ratio is determined).

Here, though, is a third argument for the irrationality of . You just work out the continued-fraction expansion. It comes out to be . Since the continued-fraction expansion of any rational number terminates, is irrational.

Consider a formal system, and construct from it a category. The objects are the well-formed formul, and the morphisms between them are proofs. This is particularly well-trodden ground. Better mathematicians than I have pointed out that in the predicate calculus, conjunction behaves like a product, implication behaves like exponentiation, and so on.

John Baez has spoken in his seminar about the particular case of typed lambda calculi, which are cartesian closed categories. When we change the tools like beta-reductions from equalities to arrows, we start talking about cartesian closed 2-categories, which hopefully will give insight into the use of lambda-calculi as models of computation, since the 2-morphisms now have the sense of a process.

For the first proof, you are (implicitly) using the identity . Equivalently, the linear transformation has as an eigenvector, and so it suffices to show that T has no eigenvector in in the sector . One then observes from parity considerations that (x,y) cannot be an eigenvector if y is odd. But then (x,y) lies in the range of , and has a smaller x coordinate, and so we can set up an infinite descent of eigenvectors, which is not possible in .

The third argument is based on the fact that is a fixed point of , and thus is an eigenvector of in the sector . This time, the determinant is 1, so we are genuinely in . The continued fraction algorithm can be viewed projectively as a dynamical system on the first quadrant of the plane which maps to when and to otherwise. This dynamical system terminates for any element of by an infinite descent; but when applied to an eigenvector of T, it maps to after three steps, giving a contradiction again.

The fourth proof has a slight typo: should be . The identity is equivalent to the assertion that has a determinant which is the negative of the determinant of . So the sequences really have to do with powers of the matrix , or equivalently the transform . I think your argument here is basically equivalent to the observation that , setting up an infinite descent for which is incompatible with being rational. (Actually, your proof is a little different, it seems to essentially use the fact that stays bounded under iteration, while (q,p) goes to infinity, but it is much the same thing.

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