How to use real analysis



Real analysis is a fundamental part of mathematics and forms the basis of very advanced mathematics such as differential calculus, functional analysis, and differential equations. This course covers the entire curriculum of the first two years of college or university. 

The first chapter of the real analysis course is the construction of real numbers. In fact, whole numbers and rational numbers are the first numbers students learn in high school. If we ask ourselves the following question: can we find a number having a square equal to 2? The answer is Yes, but this number is not a rational number. Thus certainly there is another set of numbers bigger than the set of rational numbers. This set will be called the set of real numbers. 

In order to define what is a real number, we need first to learn about the convergence of rational sequences. Let a and b be rational numbers. The absolute value of the number a is by definition an if it is positive and (-a) if it is negative. It will be denoted by |a|. We also have |a|=|-a|. The distance between a and be will be denoted by d(a,b) which is the definition d(a,b)=|a-b|. 

We define a set $\mathbb{R}$ be: $a\in \mathbb{R}$ if and only if there exists a rational sequence $(a_n)_n\subset \mathbb{Q}$ such this sequence converges to an as n goes to infinity. This set $\mathbb{R}$ is called the set of real numbers. Also, we say that the rational numbers set are dense in the real numbers set. This can also be interpreted by the following property: between any real numbers, one can find rational numbers. Let us now return to the first question: Find a number $a$ such that $a^2=2$. This number is called the square root of $a$ and denoted by $a=\sqrt{2}$. We mention that we can prove that $\sqrt{2}$ is not a rational number.

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