Really nice proof! The "clearly" thing isn't too hard to see even if you aren't aware of the "alternating series theorem": Consider the sequence a1, a2, a3...

Really nice proof! The "clearly" thing isn't too hard to see even if you aren't aware of the "alternating series theorem": Consider the sequence a1, a2, a3... with the sum being a1 - a2 + a3..., note we can write the sum as: a1 + (-a2 + a3) + (-a4 + a5) hence the sum is less than a1. We can also write it as: (a1-a2) + (a3-a4) + (a5-a6) .... hence the sum is greater than (a1-a2).