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Due: October 18th, 2019

Math 104 Assignment 6

Due to cancelled classes, I have not had time to cover most of the material which was going to form the core of this assignment, and so the selection of problems is somewhat interesting. This assignment is optional. Any points earned will be added as bonus points to your other assignments.

  1. Cauchy sequences

    1. Prove that if $(a_n)_n$ is a Cauchy sequence in some metric space $(M, d)$ with a convergent subsequence, then $(a_n)_n$ itself converges.
    2. Use the previous problem to prove that if $(K, d)$ is a compact metric space, then every Cauchy sequence in $K$ converges in $K$. (There is a proof of this fact in Rudin which uses some facts about diameters of sets to prove this. The goal of this problem is to come up with a different proof.) (Apparently I accidentally proved this in lecture.)
    3. Prove that if $(c_k)_k$ is a Cauchy sequence in some metric space $(M, d)$, then $(c_k)_k$ is bounded.

  2. Completions are unique

    Suppose that $(M, d)$ and $(X, d_X)$ are metric spaces. An isometry of $M$ into $X$ is a distance preserving function $i : M \hookrightarrow X$, i.e., one so that for all $a, b \in M$, \[d(a, b) = d_X(i(a), i(b)).\]

    1. Prove that isometries are always one-to-one.

    Suppose that $(M, d), (X, d_X)$, and $(Y, d_Y)$ are metric spaces, and that $i_X : M \hookrightarrow X$ and $i_Y : M \hookrightarrow Y$ are isometries of $M$ into $X$ and $Y$ respectively. Let us also suppose that $X$ and $Y$ are complete: that is, that every Cauchy sequence in $X$ converges to a limit in $X$, and similar for $Y$. Finally, assume that $i_X(M) = \set{i_X(a) \mid a \in M}$ is dense in $X$ (that is, $X = \overline{i_X(M)}$).

    1. Prove that there is an isometry $I : X \hookrightarrow Y$ of $X$ into $Y$ so that $i_Y(a) = I(i_X(a))$ for all $a \in M$.
    2. Prove that the isometry $I : X \hookrightarrow Y$ above is unique.
    3. Prove that if $i_Y(M)$ is dense in $Y$, then the isometry $I : X \hookrightarrow Y$ from (C) has an inverse $J : Y \hookrightarrow X$ which is also an isometry, and that $i_X(a) = J(i_Y(a))$ for all $a \in M$.

    Two metric spaces $(X, d_X)$ and $(Y, d_Y)$ are called isometric if there are isometries $I : X \hookrightarrow Y$ and $J : Y \hookrightarrow X$ which are inverses of one another. What we have now shown is that any two complete metric spaces containing $M$ as a dense manner are isometric to each other, in a way that aligns the copy of $M$ inside each. This fact is often stated as "the completion of $M$ is unique up to isometry".

  3. The real numbers as the completion of the rationals

    We will see in lecture this week that if $M$ is a metric space, there is a complete metric space $\overline{M}$ into which $M$ includes as a dense subset, and by the previous problem we know that this completion is unique. We also know that $\Q$ is a dense subset of the metric space $\R$, so $\R = \overline{\Q}$ (after applying a distance-preserving invertible function).

    Recall that earlier in the course, we merely asserted that an ordered field with the least upper bound property existed, but we didn't prove it. It is tempting to use this as our definition of what the real numbers are: to define $\R$ as $\overline{\Q}$. After making this definition, it is not so bad to show that $\R = \overline{\Q}$ has the properties of an ordered field and has the least upper bound property.

    However, there is a subtle flaw with doing this. Why is it not valid to define $\R$ as $\overline{\Q}$ in this way? What could we do to avoid this problem? (No proofs needed here; just explain the problem and give a suggestion about how to avoid it.)