Manual Zone

ShareLatex is good at swallowing the compilation errors and still producing the correct PDF. However, when locally compiling, the local errors often prevent the compilation. For example, TexStudio often fails for similar or same errors. Typing "Enter" upon encountering such errors will make the compilation proceed, regardless of such error reports.

! Missing $ inserted.
> Check empty or blank lines between the egin{equation} and end{equation} tags.

! You cant use `eqno in horizontal mode.
> Check the closing braces in the equations.

! Missing $ inserted.

                $
l.112 $forall$ n $in$ $mathbb{Z}$^{+}: $C_
                                             {Nn}$ represents the coordinato...

> Changing
$forall$ n $in$ $mathbb{Z}$^{+}: $C_ 
into
$forall$ n $in$ $mathbb{Z} ^{+}$: $C_
fixed this.