Ian Charlesworth

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

1. Suppose that $K_1, K_2 \subseteq M$ are compact. Which is the weakest assumption below which guarantees that $K_1 \cup K_2$ is compact?

   1. $K_1\cap K_2 \neq \emptyset$.
   2. $M = \R^n$ for some $n \in \N$.
   3. $K_1 \subseteq K_2$.
   4. No further assumption.

2. Which of the following is the statement that the sequence $(a_n)_n$ does not converge?

   1. $\forall L \in M \, \forall r \gt 0 \, \exists N \, \forall n \gt N, \, d(a_n, L) \geq r$.
   2. $\exists L \in M \, \exists r \gt 0 \, \forall N \, \exists n \gt N, \, d(a_n, L) \geq r$.
   3. $\forall L \in M \, \exists r \gt 0 \, \forall N \, \exists n \gt N, \, d(a_n, L) \geq r$.
   4. $\exists L \in M \, \forall r \gt 0 \, \exists N \, \forall n \gt N, \, d(a_n, L) \geq r$.

3. If $E \subseteq M$ is not compact, then no infinite open cover of $E$ has a finite subcover.

   1. True.
   2. False.

4. If $(x_n)_n$ is a sequence which takes on only finitely many distinct values, it converges.

   1. True.
   2. False.