Lecture Notes

0. Correct your mistakes in midterm 1, you don't need to submit the correction.

1. Prove that there is a sequence in $\R$, whose subsequential limit set is the entire set $\R$.

2. Prove that $\limsup(a_n+b_n) \leq \limsup(a_n) + \limsup(b_n)$, where $(a_n)$ and $(b_n)$ are bounded sequences in $\R$.

3. Give an explicit way to enumerate the set $\Z$, and then $\Z^2$.

4. Prove that the set of $\Z$-coefficient polynomials is countable.

5. Prove that the set of maps $\{f: \N \to \{0,1\}\}$ is not countable.