We will cover following topics

Completeness Of Real Line

A metric space $(X, d)$ in which every Cauchy sequence converges to an element of $X$ is called complete, where $d$ is the distance metric.
The real numbers are complete under the metric induced by the usual absolute value, and one of the standard constructions of the real numbers involves Cauchy sequences of rational numbers.
The rational numbers, on the other hand, are not complete under the usual distance metric. For example, the sequence defined by $x_{0} = 1$ , $x_{n + 1} = (x_{n} + \frac{2}{x_{n}}) / 2$ consists of rational numbers but it converges to $\sqrt{2}$ , which is irrational.

PYQs

Real number system as an ordered field

1) Suppose $R$ be the set of all real numbers and $f : R \to R$ is a function such that the following equations hold for all $x$ , $y \in R$ :

$f (x + y) = f (x) + f (y)$
$f (x y) = f (x) f (y)$

Show that that $\forall c \in R$ , either $f (x) = 0$ or $f (x) = x$ .

[2018, 20M]

2) Show that every open subset of $R$ is a countable union of disjoint open intervals.

[2013, 13M]

3) Show that a bounded infinite subset of $R$ must have a limit point.

[2009, 15M]

4) Let

$T = {\frac{1}{n}, n \in N} \cup {1 + \frac{3}{2 n}, n \in N} \cup {6 - \frac{1}{3 n}, n \in N}$ .

Find derived set $T^{'}$ of $T$ . Also find Supremum of $T$ and greatest number of $T$ .

[2008, 6M]

5) Given a positive real number $a$ and any natural number $n$ , prove that there exists one and only one positive real number $ξ$ such that $ξ^{n} = a$ .

[2007, 20M]

6) If a continuous function of $x$ satisfies the functional equation $f (x + y) = f (x) + f (y)$ , then show that $f (x) = α x$ where $α$ is a constant.

[2003, 12M]

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