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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.