The swapping indefinitely does not work. For the first iteration, if I have A in my envelope, then yes, the other envelope will either have 1/2A or 2A, meaning

The swapping indefinitely does not work. For the first iteration, if I have A in my envelope, then yes, the other envelope will either have 1/2A or 2A, meaning the expected value is indeed 5/4A. The switching argument wrongly assumes this is again the case after I switch. After I have switched, the value of the original envelope is unchanged (meaning it is A with probability 1), so I should not switch back.