Real Numbers

Prerequisites: metric spaces $\textbf{\small Definition 1.1.:}$ (Neighbourhood) Let $X$ be a metric space. A neighbourhood of a point $p$ is a set $N_r(p)$ consisting of all points $q$ such that $d(p,q) < r$. The number $r$ is called the radius of $N_r (p)$. $$ \begin{aligned} N_r(p) = \{q : d(p,q) < r \text{ where } p,q \in X\} \end{aligned} $$ $$ \begin{aligned} N_r(p) = \{q : d(p,q) < r \text{ where } p,q \in X\} \end{aligned} $$ If we are in an Euclidean metric $(\mathbb{R}^k)$ then, ...

I was freshman when I was first introduced supremum and infimum. I could not understand a thing, yet everybody seemed to understand. I felt like stupid, inadequate and also insufficient person to learn mathematics. However, reality is not like that. In truth, I was the one to blame; I hadn’t dived deep into these topics deeply enough. I do not know if I am alone or not, but I think supremum and infimum are the topics which cannot by bypassed without learning and embracing. Today I want to show an interesting fact; if a set has least-upper-bound property and bounded below, then it has infimum. But first let us define supremum and infimum. ...