What is the definition of real numbers? I only know that the rational and irrational are together called real numbers. Rational numbers can be written in the form $\frac {p}{ q}$ ,where $p \in \mathbb Z$ and $q \in \mathbb Z$.
Are there any additional information which I left? If 'yes' then please tell me the alternative definitions.How to answer it in an interview? Thank you in advance.
 A: What you need to notice first for this is that some sequences of $ \mathbb{Q} $ do not converge to a value in $ \mathbb{Q} $. In fact, for every real number, you can find a sequence of $ \mathbb{Q} $  that converges to that number (for example the different fractional series converging to $ \pi $).
The idea is to create the set of "limits" of sequences of rational numbers.
However, you can't just write $ \displaystyle\mathbb{R} = \{\lim_{n\rightarrow+\infty}u_n, u\in\mathbb{Q}^\mathbb{N}\} $, since it doesn't mean anything until you've defined $ \mathbb{R} $, because the limit itself isn't defined.
What you need is to create an equivalence relation over the "converging" subset of $ \mathbb{Q}^\mathbb{N} $ and identify the equivalence classes as elements of $ \mathbb{R} $. But first you'll need Cauchy sequences, which can be used to show that sequences have a converging behaviour but without specifying their limits.

We say that a sequence $ u $ is a Cauchy sequence iff $ \forall\varepsilon>0, \exists N, \forall n \ge N, \forall m \ge N, |u_n - u_m |<\varepsilon $
It is easy to show that if $ u $ converges in $ \mathbb{Q} $, then it is a Cauchy sequence.
Let $ A $ denote the subset of $ \mathbb{Q}^\mathbb{N} $ of Cauchy sequences.
Let $ \sim $ be a relation on $ A $, with $ \displaystyle u \sim v \iff \lim_{n\rightarrow+\infty}u_n - v_n = 0 $ (we can define this properly because $ 0 \in \mathbb{Q} $ and $ u - v \in \mathbb{Q}^\mathbb{N} $).
It is then easy to show that $ \sim $ is and equivalence relation :


*

*Reflexivity : $ \displaystyle\forall u \in A, \lim_{n\rightarrow+\infty}u_n - u_n = 0 $

*Symmetry : $ \displaystyle \forall (u, v) \in A^2, \lim_{n\rightarrow+\infty}u_n - v_n = 0 \implies \lim_{n\rightarrow+\infty}v_n - u_n = 0 $

*Transitivity : if $ u \sim v $ and $ v \sim w $, then we have $ \displaystyle \lim_{n\rightarrow+\infty}u_n - w_n = \lim_{n\rightarrow+\infty}u_n - v_n + v_n - w_n = 0 $.


Then $ \mathbb{R} = \{[u], u \in A\}$ is the set of equivalence classes of $ \sim $, and every Cauchy sequence of rational numbers defines a real number.
This construction is due to Cantor if I remember correctly. There are some other constructions using Dedekind cuts for example, but this is easy enough to understand (but no so much to use).
