Introduction to Real Analysis

In the 20th century, several challenging problems concerning real numbers have been solved. However, there remain countless simple looking problems which are not solved. For example, it is unknown whether n{!} + 1 is a square for any integer n > 7. By the time some problems are solved new problems come up, and it is the essence of the growth of the subject matter.

Starting with routine subject matter we prepare ourselves step by step to tackle the finer points while learning the art of applying the principles involved in the study of the fundamentals of the theory of real analysis.

Mathematics is the logical study of shape, size and situation. Developments in mathematics are mainly based upon the concept of numbers and the geometry of figures. Since the geometry of figures carries too many intuitive ideas, dependence on geometrical figures must be put aside. On the other hand, the theory of numbers is developed on a firm footing consistent with scientific thoughts. It is for this reason that the theory of numbers is basically adopted in all the advanced branches of mathematics.

Realizing the importance of numbers in understanding the universe, Pythagoras said as far back as about two and half thousand years that “the numbers rule the universe.” Kronecker (1823—1891) expressed the significance of numbers in the words, “God made the integers, all the rest is the work of man.” It was only in the middle of the 19th century that the importance of numbers as an independent entity was widely realized and the study of numbers was freed from geometrical intuitionalism. Three German mathematicians, K. Weierstrass (1815—1897), R. Dedekind (1831—1916), and G. Cantor (1845—19 18) mainly share the honor of being associated with the development of the theory of real numbers.

In real analysis, or the theory of real numbers, we study the development of real numbers, which follows upon several successive generalizations of the set of natural numbers. It is quite interesting to learn that when the theory of real numbers came onto the field, the theory of complex numbers (which are essentially a generalization of real numbers) was already well developed. Real analysis, however, gained a place of primary importance by way of the theory of complex numbers, as differential calculus earlier did through integral calculus. Calculus, for its complete justification, needed the support of real analysis even in the seventeenth century, but it had to wait until the middle of the nineteenth century for significant support. In fact, not only calculus but almost all branches of modem mathematics owe their strengths to the development of real analysis.

Now we proceed with developing the elements of real analysis. The symbols and various forms of notations have been explained in the appropriate places.