The error message in the format you have shown above is not readable, you should always preserve line endings when showing TeX errors.
The error on your document is
! Undefined control sequence.
l.12 \noindent 1) Show that $\sqrt{2} \in \mathbb
{R} $
? h
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.
?
Which as the help text makes clear shows, by the position of the line break in the message that \mathbb
is the problematic command that has not been defined.
TeX itself can only tell you that it is not defined, not how to define it but google for that command should quickly tell you that you need the package amsfonts
Adding
\usepackage{amsfonts}
makes the document error free, but there are rather a lot of issues with the TeX markup.
The "two letter" font commands are not recommended in latex2e (and not defined by default) it is usually better to use the latex commands, in this case \scshape
however in general you should avoid font commands in the document and especially in structural commands like \title
and \author
as the whole point of those commands is that they just have plain text with the data and the styling depends on the document class. For similar reasons you should avoid \vspace
and \noindent
and explicit numbering such as 1)
.

\documentclass[11pt]{article}
\usepackage[utf8]{inputenc}
\usepackage{amsfonts}
\makeatletter
% \@maketitle copied from article.cls, woth \ascshape added
\renewcommand\@maketitle{%
\newpage
\null
\vskip 2em%
\begin{center}%
\scshape
\let \footnote \thanks
{\LARGE \@title \par}%
\vskip 1.5em%
{\large
\lineskip .5em%
\begin{tabular}[t]{c}%
\@author
\end{tabular}\par}%
\vskip 1em%
{\large \@date}%
\end{center}%
\par
\vskip 1.5em}
\makeatother
\title{Homework 1}
\author{by Me}
\date{January 2017}
\begin{document}
\maketitle
\begin{enumerate}
\item Show that $\sqrt{2} \in \mathbb{R} $
\bigskip
We will define two sequences, $A_k$ and $B_k$.
Let $A_k = a_1a_2a_3a_4\dots a_k$, and
% You could use {A_k}^2 if you really need to stress the order
% bit it makes the 2 float off and normally the more compact
% form is used (and understood)
$A_k^2
> 2$
but decreasing the last digit, $a_k$,
by
% math mode!
$1$
will result
% as above
$A_k^2
< 2$.
\end{enumerate}
\end{document}
\usepackage{amsfonts}
in your preamble --\mathbb
is defined inamsfonts
And\sc
is an outdated command. Use\textsc{Homework 1}
etc.amsmath
, or even better,mathtools
, since you have maths formulae.\noindent
or\sc
or\vspace
in most documents nor explicit numbering like1)
you seem to be missing most of the advantages of latex to keep such formatting separate from the document content.