\documentclass[
  border=10pt,
  convert={density=300, outext=.png}
]{standalone}

% -------------------------------------------------
% Packages
% -------------------------------------------------
\usepackage{tikz}
\usepackage[T1]{fontenc}
\usepackage{ProfLycee} % French math graphics helpers
\usetikzlibrary{
  arrows.meta,
  positioning,
  calc
}

% -------------------------------------------------
% Document
% -------------------------------------------------
\begin{document}

\begin{tikzpicture}[
  x=1cm,
  y=1cm,
  xmin=-2, xmax=10,
  ymin=-2, ymax=7,
  xgrille=8, xgrilles=8,
  ygrille=7, ygrilles=8
]

  % -------------------------------------------------
  % ProfLycee window and axes labels
  % -------------------------------------------------
  % \FenetreSimpleTikz
  %   <Police=\scriptsize>{-2,-1,...,8}
  %   <Police=\scriptsize>{-2,-1,...,7}
  \FenetreTikz

  % -------------------------------------------------
  % Global parameters (overrides ProfLycee defaults)
  % -------------------------------------------------
  \def\xmin{-2}
  \def\xmax{10}
  \def\ymin{-2}
  \def\ymax{7}

  \def\xgrille{1}
  \def\ygrille{1}
  \def\xgrilles{0.5}
  \def\ygrilles{0.5}

  % -------------------------------------------------
  % Grid
  % -------------------------------------------------
  \draw[
    xstep=\xgrilles,
    ystep=\ygrilles,
    line width=0.6pt,
    lightgray!50
  ]
  (\xmin,\ymin) grid (\xmax,\ymax);

  % -------------------------------------------------
  % Axes
  % -------------------------------------------------
  \draw[
    line width=1.5pt,
    ->,
    darkgray,
    >=latex
  ]
  (\xmin,0) -- (\xmax,0);

  \draw[
    line width=1.5pt,
    ->,
    darkgray,
    >=latex
  ]
  (0,\ymin) -- (0,\ymax);

  % -------------------------------------------------
  % Tick marks
  % -------------------------------------------------
  \foreach \x in {0,1,...,9} {
    \draw[darkgray, line width=1.5pt]
      (\x,4pt) -- (\x,-4pt);
  }

  \foreach \y in {0,1,...,7} {
    \draw[darkgray, line width=1.5pt]
      (4pt,\y) -- (-4pt,\y);
  }

  % -------------------------------------------------
  % Function definition
  % f(x) = sqrt(5x) + 1
  % -------------------------------------------------
  \def\f{sqrt(5*\x) + 1}

  % -------------------------------------------------
  % Cobweb / recurrence diagram
  % -------------------------------------------------
  \ToileRecurrence[
    Fct={\f},
    No=0,
    Uno=1,
    Nb=5,
    DecalLabel=4pt
  ]

  % -------------------------------------------------
  % Function curve y = f(x)
  % -------------------------------------------------
  \draw[
    very thick,
    blue,
    domain=0:8,
    samples=250
  ]
  plot (\x,{\f});

  % -------------------------------------------------
  % Identity line y = x
  % -------------------------------------------------
  \draw[
    very thick,
    domain=0:8,
    samples=2
  ]
  plot (\x,\x);

  \node at (0.75,-1.5) {$U_{0}=1$};
  \node[text width=4cm, align=left] at (2.2,-0.9) {$U_{n+1}=f(U_n)=\sqrt{5U_n}+1$};
  \node at (2.4,6.2) {$f(x)=\sqrt{5x}+1$};
  \node[text width=4cm, align=left] at (9,5) {$U_{1} \approx 3.236$};
  %\node[text width=4cm, align=left,red] at (9,5) {$f(1)=3.236$};
  \node[text width=4cm, align=left] at (9,4) {$U_{2} \approx 5.022$};
  %\node[text width=4cm, align=left,red] at (9,3.5) {$f(3.236)= 5.022$};
  \node[text width=4cm, align=left] at (9,3) {$U_{3}\approx 6.011$};
  %\node[text width=4cm, align=left,red] at (9,2) {$f(5.022)= 6.011$};
  \node[text width=4cm, align=left] at (9,2) {$U_{4} \approx 6.482$};
  \node[text width=4cm, align=left] at (6.5,-1.2) {
    \boxed{ \displaystyle \lim U_n = \frac{7+3\sqrt{5}}{2} \approx 6.854 }.
  };
  %\node at (7,4) {$U_{1}=sqrt(sqrt(5)+1)+1$};
  % sqrt(5*1)+1 = 3.23606797749979
  % sqrt(5*(sqrt(5)+1))+1 = 5.022479320953552
  % sqrt(5*(sqrt(5*(sqrt(5)+1))+1))+1 = 6.011227055798585
  % sqrt(5*(sqrt(5*(sqrt(5*(sqrt(5)+1))+1))+1))+1 = 6.482347606545295
\end{tikzpicture}

\end{document}