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

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

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

\section{DEFINITIONS}

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

\begin{equation}
	I_{1} \coloneqq \int_{u_{0}}^{u} S(t) dt, \quad t = u, \quad u > 0
\end{equation}
Where $u, u_{0} > 0, \quad u, u_{0} \in \mathbb{R}$, such that $S : (u, \pi) \to \mathbb{R}$.

\section{LEMMAS}

\subsection{FIRST LEMMA}

\begin{defn}We assume the value $B_{0} \in \mathbb{R}, \quad \pi > B_{0} > 0$. \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 assume that:
\begin{equation}
    \lim_{B_{n} \to \infty} = B_{n} \to I_{1}, \quad
\end{equation}
If and only if: $B_{0} > 0, \quad B_{n} \in \mathbb{R}$.
\end{proof}

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