Ian Charlesworth

Let $(\mathcal{M}, d)$ be a metric space.

1. If every sequence in $\mathcal{M}$ has a convergent subsequence, then $\mathcal{M}$ is complete.
   1. True
   2. False
2. If $\mathcal M$ is complete, then every sequence in $\mathcal M$ has a convergent subsequence.
   1. True
   2. False

3. Suppose $(x_n)_n$ is a sequence in $\R$, and $X = \set{x_n \mid n \in \N}$. We wish to prove that if $x \in X'$ there is a subsequence converging to $x$. Is the following proof valid?

   For each $k \in \N$, since $x$ is a limit point of $X$, there is some $y \in X$ so that $d(x, y) \lt \frac1k$. Since $y$ occurs in the sequence, there is some index $n_k$ so that $x_{n_k} = y$. Now if we take $(x_{n_k})_k$, this converges to $x$ by construction (indeed, given $\epsilon \gt 0$, if $k \gt \frac1\epsilon$ we have $d(x_{n_k}, x) \lt \frac1k \lt \epsilon$). Thus there is a subsequence converging to $x$.

   1. Yes
   2. No
4. Suppose $F \subseteq \mathcal M$ and $(x_n)_n$ is a sequence in $F$ which converges. Then the limit of the sequence is a limit point of $F$.
   1. True
   2. True as long as $F' \neq \emptyset$
   3. False
5. Are these polls useful/helpful/worth doing?
   1. Yes
   2. No
   3. Meh