Let $A_i$ be a subset of a metric space for each $i\in \mathbb{N}$.
- Let $B_n := \bigcup_{i=1}^n A_i$. Prove (for any) $n \in \mathbb{N}$ that $\overline{B_n} = \bigcup_{i=1}^n \overline{A_i}$.
- If $B = \bigcup_{i=1}^\infty A_i$, prove that $\overline{B} \supseteq \bigcup_{i=1}^\infty \overline{A_i}$. Give an example to show that this containment might be proper.
If $A_i$ is closed then $A_i = \bar A_i$, but I'm stuck as to how to prove $B=\bar B_n$. If I prove the first statement for when $A_i$ is closed does that mean it is also true for when $A_i$ is open because I can construct a closed set containing $A_i$?
For the example, would constructing a sequence of closed segments between $0$ and $1$ that gets arbitrarily close to $1$ and taking the union of the segments be considered a proper containment?