% AUTHOR: Amlal El Mahrouss
% PURPOSE: Lemmas for Integrals.

\documentclass[11pt, a4paper]{article}
\usepackage{graphicx}
\usepackage{listings}
\usepackage{xcolor}
\usepackage{hyperref}
\usepackage{mathtools}
\usepackage{amsfonts}
\usepackage[margin=0.5in,top=1in,bottom=1in]{geometry}

\newenvironment{proof}{\paragraph{PROOF:}}{\hfill$\square$}

\newtheorem{thm}{LEMMA}[section]
\newtheorem{defn}[thm]{DEFINITION}

\title{Lemmas for Integrals.}
\author{Amlal El Mahrouss\\amlal@nekernel.org}
\date{February 2026}

\begin{document}
\maketitle

\begin{abstract}
\begin{center}
    The paper covers several lemmas and definitions that we will define to help us connect the dots for series in analysis.
\end{center}
\end{abstract}

\begin{center}
  \rule[0.01cm]{17cm}{0.01cm}
\end{center}

\section{DEFINITIONS}

We assume that $u \in \mathbb{R}, \quad \forall u, \pi > u > 0$

\begin{equation}
	I_{n} \coloneqq \int_{u_{n}}^{u} X(t) dt, \quad \forall n \geq 1, \quad n \in \mathbb{Z}
\end{equation}
Where $u, u_{0} > 0$ for $X(u) \in \mathbb{R}$ where $X(u)$ is a function in $\mathbb{R}$.

\section{LEMMAS}

\subsection{FIRST LEMMA}

\begin{defn}We assume that $B_{0} \in \mathbb{R}, \quad \pi > B_{0} > 0$. \end{defn}
\begin{defn}We assume $\forall n \geq 1, \quad n \in \mathbb{Z}$. \end{defn}
\begin{thm}For all variables of $B \to B_{n}$ such that $0 < B_{n} < \pi$, the variable $B_{n}$ is defined in $I_{1}$.\end{thm}

\begin{proof}
We assert that:
\begin{equation}
    \lim_{n \to +\infty} f_{n} \coloneqq B_{n} \to I_{n}, \quad
\end{equation}
If and only if: $B_{0} > 0, \quad B_{n} \in \mathbb{R}$.
\end{proof}

\end{document}
