\documentclass[11pt]{article}	% you can replace with 10pt (smaller) or 12pt (larger)


% Packages %

\usepackage{geometry}
%\geometry{a4paper}
%\geometry{margin=1in}		%  remove "%" symbol in front of \geometry to make 1 inch margins

\usepackage{amsmath}	% this package allows you to use math symbols, you can look up what other packages do via google
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{latexsym}
\usepackage{titlesec}
\usepackage{graphicx}	% this package lets you insert images into your file


%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\begin{center}	% centers
\Large{Sample Homework}	% Large makes the font larger, put title inside { }
\end{center}
\begin{center}
Victoria Kala \\
Math 117 \\
August 2, 2016
\end{center}

\begin{enumerate}	% this makes a numbered list

%%%%%

\item Statement of problem 1

\begin{enumerate}	% this makes a numbered list within the numbered list, starts with (a) by default

	\item Statement of part (a) of problem
		\begin{proof}
		
		Here I type the proof of my problem. Notice that LaTeX will automatically place ``Proof" at the beginning of the proof and an open square at the end of the proof.
	
		\end{proof}
	
	\item Statement of (b) of problem
	
		\begin{proof}[Solution]
		 Maybe I am not proving something but want to state a solution. I can still use the``Proof" environment, just use `[ ]' symbols to rename ``Proof" to ``Solution". \\ % the double \ with an extra return will automatically create a new paragraph
	 
	 	Notice that I must use the ` symbol to make left apostrophe or use it twice to make left quotation marks, otherwise I only get right apostrophes or quotation marks like so:
	 
	  	"look at these weird 'quotation' marks". 	% didn't use \\ here so I only get a new line, not a new paragraph
		\end{proof}
	
\end{enumerate} % ends the numbered list we were using earlier

%%%%%

\item[200.] Statement of problem 200	% You can rename items with [ ] 

	\begin{proof}
	
	Use the dollar symbol to enter math mode: $5x^2 + 100 \frac{e}{\pi} + \ln{500}$.\\
	
	Use square brackets and backslash to enter math mode, but on a centered new line:
	\[ \forall \varepsilon > 0, \exists ~ n \in \mathbb{N} ~ \text{s.t.} ~ n, m \geq N \Rightarrow |a_n - a_m| < \varepsilon. \]
	% the ~ symbol puts spaces inbetween math characters
	
	Use equation environment to write an equation on a centered new line with a number next to it:
		\begin{equation}
		\frac{d}{dx} \arctan{x} = \frac{1}{1+x^2}
		\end{equation}
	
	You can also rename equations:
		\begin{equation}\tag{105}
		\lim_{x \to 0} e^x = 1
		\end{equation}
	
	\end{proof}		
	
%%%%%

\item Statement of problem 2

	\begin{proof}[Solution]
	Maybe I have a set of steps I want to show, I can use the align environment:
	\begin{align*}
		\int_0^{\pi} \sin{x} dx & = - \cos{x} \Big|_0^{\pi} \\
			& = - \cos{\pi} - (-\cos{0}) \\
			& = - (-1) - (-1) \\
			& = 1 + 1 \\
			& = 2
	\end{align*}
	
	\end{proof}



\end{enumerate}	% ends the numbered list we made earlier

\end{document}