A) 1 doesn't imply 2 if we take as a fact infinite divisibility of the bitcoin (which is apparently the case, so I do not see why awards cannot be continually halved)
It is not the case. Storing fractional numbers with infinite precision would require a computer with infinite memory, which is of course impossible (there are systems for storing numbers with "arbitrary" precision, but they still have a finite limit determined by the amount of memory available). In the case of Bitcoin, fractional units are only stored to 8 decimal places. The block reward is always rounded down (never up) when 8 decimal places aren't enough, which will first happen at the 10th block reward halving (sometime around the year 2048) when the reward drops from
BTC0.09765625 to
BTC0.04882812 (rounded down from
BTC0.048828125). Eventually (the 33rd halving, around the year 2140), the block reward will drop from
BTC0.00000001 to
BTC0.00000000, no more bitcoins will ever be generated, and the currency supply will be fixed at exactly
BTC20,999,999.97690000 (the series of block reward halvings theoretically converges on exactly 21 million, but it does not actually reach this number due to the aforementioned rounding errors).
B) Supposing that the # of bitcoins in circulation tops out at N= 21 million, at that time which implies enough halving periods have passed so that no new bitcoins are mined, what incentive does anyone have at that time to verify the blockchain anymore? Currently the reward of bitcoins is clearly the reason why transactions are verified at all.
Miners receive transaction fees in addition to the block reward. As the block reward gets smaller and smaller, these fees will make up a larger and larger proportion of the total payment received by miners, resulting in a smooth transition to the eventual "transaction fee only" model.
