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The presentation of real analysis in advanced texts usually starts with simple proofs in elementary set theory, a clean definition of the concept of function, and an introduction to the natural numbers and the important proof technique of mathematical induction.
Then the real numbers are either introduced axiomatically, or they are constructed from sequences of rational numbers. Initial consequences are derived, most importantly the properties of the absolute valueIn mathematics, the absolute value (or modulus of a number is that number without a negative sign. So, for example, 3 is the absolute value of both 3 and −3. Definition It can be defined as follows: For any real number a the absolute value of a deno such as the triangle inequalityIn mathematics, the triangle inequality is a statement which states roughly that the distance from A to B to C is never shorter than going directly from A to C. The triangle inequality is a theorem in spaces such as the real numbers, Euclidean space, Lp s and Bernoulli's inequalityBernoulli's inequality in real analysis states that : for every integer n ≥ 0 and every real number x ≥ −1. If n ≥ 0 is even, then the inequality is valid for all real numbers x''. The strict version of the inequality reads : for every inte.
The concept of convergence, central to analysis, is introduced via limitsIn mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets "close" to either some point, or infinity; or the behavior of a sequence's elements, as their index approaches infinity. Limits are used in calculu of sequences. Several laws governing the limiting process can be derived, and several limits can be computed. Infinite series, which are special sequences, are also studied at this point. Power seriesIn mathematics, a power series (in one variable) is an infinite series of the form :: where the coefficients a the center a and the argument x are usually real or complex numbers. These series usually arise as the Taylor series of some known function; the serve to cleanly define several central functions, such as the exponential functionThe exponential function is one of the most important functions in mathematics. It is written as exp x or e x where e is the base of the natural logarithm. As a function of the real variable x the graph of e x is always positive (above the x axis) and inc and the trigonometric functionIn mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. They may be defined as ratios of two sides of a right triangle containing the angle, or, more generally, as ratios ofs. Various important types of subsets of the real numbers, such as open sets, closed sets, compact sets and their properties are introduced next.
The concept of continuity may now be defined via limits. One can show that the sum, product, composition and quotient of continuous functions is continuous, and the important intermediate value theorem is proven. The notion of derivative may be introduced as a particular limiting process, and the familiar differentiation rules from calculus can be proven rigorously. A central theorem here is the mean value theorem.
Then one can do integration ( Riemann and Lebesgue) and prove the fundamental theorem of calculus, typically using the mean value theorem.
At this point, it is useful to study the notions of continuity and convergence in a more abstract setting, in order to later consider spaces of functions. This is done in point set topology and using metric spaces. Concepts such as compactness, completeness, connectedness, uniform continuity, separability, Lipschitz maps, contractive maps are defined and investigated.
We can take limits of functions and attempt to change the orders of integrals, derivatives and limits. The notion of uniform convergence is important in this context. Here, it is useful to have a rudimentary knowledge of normed vector spaces and inner product spaces. Taylor series can also be introduced here.
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