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\date{Due Thursday, 1/31, reading exercises due earlier}
\title{Math 104 Homework 1 (Vaintrob)}
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\section{Exercise 4.1, 4.2} left column: (a, c, e, g, i, k, m, o, q, s, u, w)

\section{Exercise 4.1, 4.2} left column

\section{Exercises 4.6, 4.7}

\section{Exercise 5.3} left column

\section{6.1}

\section{Challenge problem} You can choose to do this instead of any three of the above.

Say $(s_1,s_2,\dots)$ is a sequence of distinct points in the interval $[0,1]$. Show that there is a ``subsequence'' $(t_1, t_2, t_3,\dots)$ such that each $t_i$ is one of the $s_j$ and such that the $t_i$ have a limit.

Hint: {\bf part 1}: split the interval into two pieces $[0,1/2]$ and $[1/2, 1].$ Since there are infinitely many distinct points $s_i,$ there must either be infinitely many points with first binary digit $0.0$ or infinitely many points with first binary digit $0.1$. Make a similar argument to get infinitely many elements with the same first two digits, then three digits, etc. This lets you generate all (infinitely many) digits of a binary number, $b$ between $0$ and $1$.

{\bf Part 2}. Now find some subsequence $(t_1,t_2,\dots)$ such that $t_1$ has the same first digit as $b$, such that $t_2$ has the correct first two digits as $b$, etc. Finally show that the limit of $t_1,t_2,\dots$ is $b$.



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