**Lemma:** Let be a sequence such that converges, but does not converge absolutely. For each , define and . Then both and diverge to infinity.

Essentially, the idea is to consider the sums of the positive and negative terms separately. If both of these series converged, then the original series would be absolutely convergent, which contradicts the hypothesis that the series converges only conditionally. If one of the series is convergent and the other divergent, then the sum of the two series must diverge (basically, something infinite plus something finite must be infinite). We get a proof of the lemma by making this rigorous.

Once the lemma is proved, we can prove the rearrangement theorem.

**Theorem:** Suppose that converges, but not absolutely, and let be any extended real number (i.e. any real number, or positive or negative infinity). Then there is a permutation of such that .

Again, I’m not going to try to be rigorous here, but the basic idea is that we can first arrange the positive terms in decreasing order, and the negative terms in increasing order. Since the series converges conditionally, the sequence must tend to zero. This means that the sequences of positive and negative terms must also tend to zero. Now, add positive terms together until you get something just bigger than , then add negative terms to that until you get something just less than , then add positive terms again until you get something just bigger than , and so on. Since the amounts that you are adding and subtracting are getting smaller, the error shrinks as we repeat this process over and over again. In the limit, we get exactly . But was an arbitrary constant that we fixed at start, which gives the proof.

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