I like the idea of only having 21 million bitcoins.
Bitcoins get lost all the time, lost private keys.
In time we will have fewer and fewer bitcoins, currently it is thought 3-4 million bitcoins are already lost. This makes me sad.
I propose a redistribution method:
>If an address did not have any activity in let's say the last 100 years the address is considered lost and the bitcoins are free to be mined in the next block.
I think that is a horrendous idea. Without a clear definition of the coin that is being lost and a proven method of counting lost coins, coin recirculation would be prone to a bigger problem.
Besides, for the sake of keeping the roughly 21 million bitcoins intact, I think that is a lame reason that has no urgency. There are intended coins burned by OP_RETURN. So that won't happen. If we truly want to address the legitimate users who forgot, missing, corrupt private keys, or has any other genuine cause. That might be the sensible motive. But still, it is a tricky one dues to the nature of how bitcoin essentially works.
(I am not aware of any other method of "proving" you still have access to an address).
Signed messages. Take a look at
Digital Signatures for further information.