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I would like to correct the attached script that I was lucky enough to have it thanks to this Post

documentclass[11pt,draft]book
usepackage[T1]fontenc
usepackage[utf8]inputenc
usepackage[french]babel
usepackageamsmath
usepackageamssymb
usepackagemathrsfs
usepackageamsfonts
usepackagemathtools
usepackagentheorem
usepackage[most]tcolorbox
usepackage[nosolutionfiles]answers % for immediate answers (not stored in a file)
theoremstyleplain
theoremindent=40pt
theoremheaderfontnormalfontbfserieshspace-theoremindent
newtheoremtheoremThéorème
newtheoremdefinitionDéfinition
newtheoremproofDémonstration
usetikzlibrarydecorations.pathmorphing
tcbuselibraryskins
tcolorboxenvironmenttheorem
 blanker,
 breakable,
 before skip=topsep,
 after skip=topsep,
 borderline west=1pt20ptdouble, shorten <=12pt
 
tcolorboxenvironmentproposition
 blanker,
 breakable,
 before skip=topsep,
 after skip=topsep,
 borderline west=1pt20ptdouble, shorten <=12pt
 
tcolorboxenvironmentdefinition
 blanker,
 breakable,
 before skip=topsep,
 after skip=topsep,
 borderline west=1pt20ptdecoration=zigzag,amplitude=2pt,pre=moveto,pre length=12pt,decorate
 
newtheoremexoExercice
NewassociationcorrectionSolnmycor
NewassociationindicationIndimyind
renewcommandSolnlabel[1]bf emphCorrection #1
renewcommandIndilabel[1]bf emphIndication #1
newcommandflag[1]
newtheoremquestionQuestion
defTinyfontsize4pt4ptselectfont
newcommand*eqdefensuremathoversetmathclaptextTiny def=
newcommandnorm[1]leftlVert#1rightrVert
begindocument
hrule
begincenter
textbftextsfLarge
 Exercices % TITRE

endcenter
hrule
vspace*1cm
Opensolutionfilemycor[ficcorex]
Opensolutionfilemyind[ficind]

beginexo
Soient $(X,d)$ un espace métrique et $Asubset X$, $Aneqemptyset$. On pose:
 [
 d(x, A)=inf d(x,a) ;, ain A.
 ]
 Qu'on appelle la textit distance du point $x$ à l'ensemble $A$.
 On remarque que $d(x, A)$ est bien définie et $geqslant 0$. De plus, $d(x,A)=0$ si $xin A$.
 Montrer que
beginenumerate
 item $xmapsto d(x,A)$ est $1$-lipschitzienne.
 item $d(x,A)=d(x,overline A)$.
 item $d(x, A)=0Longleftrightarrow xin overline A$.
 endenumerate
endexo
begincorrection
beginenumerate
 item Soient $x, yin A$. Pour tout $ain A$ on a
 [
 -d(x,y)+d(y,a)leqslant d(x,a)leqslant d(y,a)+d(x,y).
 ] Par passage à l'inf, on trouve
 $-d(x,y)+d(y, A)leqslant d(x,A)leqslant d(y,A)+d(x,y)$, d'où $|d(x, A)-d(y,A)|leqslant d(x,y)$.
 item Clairement, si $Asubset B$, alors $d(x, A)geqslant d(x, B)$. En particulier, $d(x, A)geqslant d(x,overline A)$. Soit $varepsilon>0$. Il existe un $binoverline A$ tel que $d(x,b)<d(x,overline A)+varepsilon/2$. Il existe une suite $(a_n)subset A$ telle que $a_nto b$. Comme $ymapsto d(x,y)$ est continue sur $X$, on trouve $d(x, a_n)to d(x,b)$. Il existe donc un $n_0$ tel que $d(a_n_0,b)<varepsilon/2$. Il s'ensuit que
 $$d(x, A)leqslant d(x, a_n_0)leqslant d(x,b)+d(b, a_n_0)<d(x,overline A)+varepsilon.$$
 $varepsilon$ étant arbitraire, on trouve $d(x, A)leqslant d(x,overline A)$.\
 item "$Longrightarrow$" Si $d(x, A)=0$, alors pour tout $varepsilon>0$ il existe un $ain A$ tel que $d(x,a)<varepsilon$. Pour $varepsilon=1/(n+1)$, on trouve une suite $(a_n)subset A$ telle que $d(x,a_n)<1/(n+1)$. On a donc $a_nto x$, et par conséquent $xinoverline A$.\
 Si $xinoverline A$, alors $0=d(x,overline A)=d(x,A)$.
endenumerate
 endcorrection
enddocument

Here is an error message

Overfull hbox (6.10593pt too wide) in paragraph at lines 87--88 

How to remove the small black rectangle that appears at the end of the correction?

Such problems should be tackled when the text is in final form.

The black box, together with the warning about Overfull hbox, signals that TeX was unable to find a suitable break point. There are several actions that can be tried.

My first advice is to avoid an entire correction to consist of an enumerate environment, which gives an unbalanced look to the page. Another general advice is to load the microtype package (although in this case it doesn't help).

In your case, a 6pt overfull means that the geqslant symbol is almost completely in the margin, so this should be fixed.

In the code that follows, I show the original, then some possible fixes. In the first fix I keep enumerate and push the geqslant symbol to the next line: in a real emergency, this might do, once in a document. The best fix, in any case, is to just omit “Clairement”.

Note the thickmuskip trick where I just add some shrinkability to it: again, once in a document, a reduced space around a relation symbol is not so bad. Don't do a global setting: this could have a very bad impact on formula spacing.

documentclass[11pt,draft,a4paper]book
usepackage[T1]fontenc
usepackage[utf8]inputenc
usepackage[french]babel
usepackageamsmath
usepackageamssymb

begindocument

subsubsection*Original within textttenumerate

beginenumeratesetcounterenumi1
item Clairement, si $Asubset B$, alors $d(x, A)geqslant d(x, B)$. 
En particulier,
%%%% the problem!
$d(x, A)geqslant d(x,overline A)$.
%%%%
Soit $varepsilon>0$. Il existe un $binoverline A$ tel que 
$d(x,b)<d(x,overline A)+varepsilon/2$. Il existe une suite $(a_n)subset A$ telle 
que $a_nto b$. Comme $ymapsto d(x,y)$ est continue sur $X$, on trouve $d(x, a_n)to d(x,b)$. 
Il existe donc un $n_0$ tel que $d(a_n_0,b)<varepsilon/2$.
endenumerate

subsubsection*Original with textttchar`\linebreak

beginenumeratesetcounterenumi1
item Clairement, si $Asubset B$, alors $d(x, A)geqslant d(x, B)$. 
En particulier,
%%%% the problem!
$d(x, A)linebreak[0]geqslant d(x,overline A)$.
%%%%
Soit $varepsilon>0$. Il existe un $binoverline A$ tel que 
$d(x,b)<d(x,overline A)+varepsilon/2$. Il existe une suite $(a_n)subset A$ telle 
que $a_nto b$. Comme $ymapsto d(x,y)$ est continue sur $X$, on trouve $d(x, a_n)to d(x,b)$. 
Il existe donc un $n_0$ tel que $d(a_n_0,b)<varepsilon/2$.
endenumerate

subsubsection*Without textttenumerate

2. Clairement, si $Asubset B$, alors $d(x, A)geqslant d(x, B)$. 
En particulier,
%%%% the problem!
$d(x, A)geqslant d(x,overline A)$.
%%%%
Soit $varepsilon>0$. Il existe un $binoverline A$ tel que 
$d(x,b)<d(x,overline A)+varepsilon/2$. Il existe une suite $(a_n)subset A$ telle 
que $a_nto b$. Comme $ymapsto d(x,y)$ est continue sur $X$, on trouve $d(x, a_n)to d(x,b)$. 
Il existe donc un $n_0$ tel que $d(a_n_0,b)<varepsilon/2$.

subsubsection*Locally reduce textttchar`\thickmuskip

2. Clairement, si $Asubset B$, alors $d(x, A)geqslant d(x, B)$. 
En particulier,
%%%% the problem!
$thickmuskip=muexprthickmuskip + 0mu plus 0mu minus 2murelax 
d(x, A)geqslant d(x,overline A)$.
%%%%
Soit $varepsilon>0$. Il existe un $binoverline A$ tel que 
$d(x,b)<d(x,overline A)+varepsilon/2$. Il existe une suite $(a_n)subset A$ telle 
que $a_nto b$. Comme $ymapsto d(x,y)$ est continue sur $X$, on trouve $d(x, a_n)to d(x,b)$. 
Il existe donc un $n_0$ tel que $d(a_n_0,b)<varepsilon/2$.

subsubsection*Rewording

2. Si $Asubset B$, alors $d(x, A)geqslant d(x, B)$. 
En particulier,
%%%% the problem!
$d(x, A)geqslant d(x,overline A)$.
%%%%
Soit $varepsilon>0$. Il existe un $binoverline A$ tel que 
$d(x,b)<d(x,overline A)+varepsilon/2$. Il existe une suite $(a_n)subset A$ telle 
que $a_nto b$. Comme $ymapsto d(x,y)$ est continue sur $X$, on trouve $d(x, a_n)to d(x,b)$. 
Il existe donc un $n_0$ tel que $d(a_n_0,b)<varepsilon/2$.

enddocument

The title of your posting is a bit misleading: the black rectangle -- it's called a "slug" in typography -- occurs at the end of an overfull line, not at the end of an environment. The slugs are being inserted because you specified the draft document class option.

• The worst way to get rid of the slugs is to remove the draft document class option. While it gets rid of the slugs, it does nothing to fix the issues that cause the occurrence of overfull lines.




• The best way to get rid of a slug is to rewrite parts or all of the paragraph in question, so that LaTeX can find other, better line breaks.




• If rephrasing sentences and paragraphs isn't an option, I suggest you change the value of thickmuskip -- a parameter that governs how much whitespace is inserted to the left and right of "relational" operators such as subset and geqslant. The default value of thickmuskip is 5mu; setting it to 3mu gets rid of the overfull line and hence gets rid of the slug.

The following is a much-simplified version of your code, to allow us to focus on just the enumerated item where the slug occurs. I suggest that if you're choose the reduce the value of thickmuskip, you reduce the value of medmuskip (which governs the spacing around binary operators) as well.

documentclass[11pt,draft]book
usepackage[T1]fontenc
usepackage[utf8]inputenc
usepackage[french]babel
usepackagemathtools,amssymb,mathrsfs
usepackage[nosolutionfiles]answers % for immediate answers (not stored in a file)

usepackagentheorem
theoremstyleplain
theoremindent=40pt
theoremheaderfontnormalfontbfserieshspace-theoremindent
NewassociationcorrectionSolnmycor
renewcommandSolnlabel[1]bf emphCorrection #1
begindocument

hrule
begincenter% TITRE
textbftextsfLarge Exercices
endcenter
hrule
vspace*1cm

begincorrection
beginenumerate
item Soient $x, yin A$. Pour tout $ain A$ on a dots

item Clairement, si $Asubset B$, alors $d(x, A)geqslant d(x, B)$. En particulier, $d(x, A)geqslant d(x,overline A)$. Soit $varepsilon>0$. Il existe un $binoverline A$ tel que $d(x,b)<d(x,overline A)+varepsilon/2$. Il existe une suite $(a_n)subset A$ telle que $a_nto b$. dots

item begingroup % localize scope of next two instructions
thickmuskip=3mu % default value: 5mu
medmuskip=3mu % default value: 4mu
Clairement, si $Asubset B$, alors $d(x, A)geqslant d(x, B)$. En particulier, $d(x, A)geqslant d(x,overline A)$. Soit $varepsilon>0$. Il existe un $binoverline A$ tel que $d(x,b)<d(x,overline A)+varepsilon/2$. Il existe une suite $(a_n)subset A$ telle que $a_nto b$. dots 
endgroup

item Clairement, si $Asubset B$, alors $d(x, A)geqslant d(x, B)$. En particulier, $d(x, A)geqslant d(x,overline A)$. Soit $varepsilon>0$. Il existe un $binoverline A$ tel que $d(x,b)<d(x,overline A)+varepsilon/2$. Il existe une suite $(a_n)subset A$ telle que $a_nto b$. dots
endenumerate
endcorrection
enddocument