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Exercise:
[A 2004 Notices article on APOS theory of teaching claims that of 13
 students they asked, only 1 got this, and after heavy prodding.
 My conclusion is that the student had been taught poorly,
 or the question was inappropriate -- it's a trivial problem.]
[I'm writing up this exercise as a reaction, to show how stupid these
 authors are.]

Is $\bigcup_{k\in \bN} \cP(\set{1,2,\dots,k})=\cP(\bN)$?
(prove or disprove)
Hint:
- $A \in \cup B_i$ iff $A \in B_i$ for -some- $i$
- think of some specific element of $\cP(\bN)$.
  Is it in $\cP(\set{1,2,\dots,k})$ for some $k$?
  (which $k$?)
  The first or second element you think of should answer this.
Answer:
$\bN \in \cP(\bN)$, but it's not in any $\cP(\set{1,2,\dots,k)}$,
so they're not equal. (Perhaps easier: the even numbers are a subset,
but they're not in any finite subset.)

That is:
- if you understand the concepts
  (union, power set)
- and think of the most -basic- example,
- and you're done

Part II:
Okay, so they're not equal.
Describe what $\bigcup_{k\in \bN} \cP(\set{1,2,\dots,k})$ is.
A: It's all -finite- subsets of $\bN$.
