So I had this thought that I was trying to prove as an exercise
Let $\mathbb{R}$ be the set of real numbers and let $\mathbb{S}$ be the set of all possible rearrangements of the alternating harmonic series. Prove that $|\mathbb{R}| < |\mathbb{S}|$
I thought I had a proof of this, but I then posted it to Reddit /r/math only to be downvoted and told the proof was wrong. The only comment I received was to "look at it from the other direction", but that confused me.
Here is my proof:
Two sets have the same cardinality iff there exists a bijection between them.
From the rearrangement theorem we can show that a the alternating harmonic series can converge to any real number via the following algorithm:
Start with $1$, if this is larger than the target number add the next negative term, otherwise add the next positive term. We create a mapping from the created rearrangement to the limit of this rearrangement. Notice that this maps to all real numbers.
Now take one of the series that we had, and switch the first two terms. This is a new rearrangement since it does not begin with $1$, so it should be mapped to a new real number. However all real numbers have already had a rearrangement mapped to them. As such we have two rearrangements pointing to a single real number, which means that our mapping is not a bijection.
As such there must be more rearrangements than real numbers.
Now I am not sure where my proof went wrong, so any help would be appreciated!