%                              ▄▄        
% ▀████▄     ▄███▀         ██  ███        
%   ████    ████           ██   ██        
%   █ ██   ▄█ ██  ▄█▀██▄ ██████ ███████▄  
%   █  ██  █▀ ██ ██   ██   ██   ██    ██  
%   █  ██▄█▀  ██  ▄█████   ██   ██    ██  
%   █  ▀██▀   ██ ██   ██   ██   ██    ██  
% ▄███▄ ▀▀  ▄████▄████▀██▄ ▀███████  ████▄
                                        

\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\I}{\mathbb{I}}
\newcommand{\E}{\mathbb{E}}
\newcommand{\D}{\mathbb{D}}
\newcommand{\8}{\infty}
\newcommand{\dx}{\mbox{d}}
\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}
\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil}
\newcommand{\hastoeq}{\overset{!}{=}}
\newcommand{\Leftrightarrowequiv}{\overset{\cdot}{\Leftrightarrow}}
\newcommand{\Longleftrightarrowequiv}{\overset{\cdot}{\Longleftrightarrow}}
\newcommand{\Rightarrowequiv}{\overset{\cdot}{\Rightarrow}}
\newcommand{\Leftarrowequiv}{\overset{\cdot}{\Leftarrow}}
\newcommand{\defAs}{\overset{\text{def}}{=}}
\newcommand{\der}[1]{\frac{\mbox{d}}{\mbox{d}#1}}
\newcommand{\dern}[2]{\frac{\mbox{d}^{#2}}{\mbox{d}^{#2}#1}}
\newcommand{\defEquiv}{\overset{\text{def}}{\Leftrightarrow}}
\newcommand{\defImplies}{\overset{\text{def}}{\Rightarrow}}
\newcommand{\divides}{\hspace{0.25em}|\hspace{0.25em}}
\newcommand{\mbeq}{\overset{!}{=}}
\newcommand{\lcm}{\text{lcm}}
\newcommand{\limit}[2]{\lim_{#1 \rightarrow #2}}
\newcommand{\seq}[1]{(#1_n)_{n \geq 1}}
\newcommand{\ser}[2]{\sum_{k = 1}^{#2} #1_k}
\newcommand{\limni}{\limit{n}{\infty}}
\newcommand{\liminfni}{\liminf_{n \rightarrow \infty}}
\newcommand{\limsupni}{\limsup_{n \rightarrow \infty}}
\DeclareMathOperator\arctanh{arctanh}
\DeclareMathOperator\arccosh{arccosh}
\DeclareMathOperator\arcsinh{arcsinh}
