The person would either die first or be financially crippled by the momumental posting charges.

You would never know because if there were an infinite number of PO boxes, then to place 5 items in each box would continue infinitely. You cannot beat infinity!

Infinity is not a number, it is a process. You simply cannot do this sort of arithmetic on infinite items. It is not mathematically meaningful: multiplying by infinity is the same as dividing by zero.

Personally, I feel there would be 5 times as much mail as boxes ... however, the only known consistant mathematical way of treating infinite sets is to use the methods of Cantor. He said that counting only makes sense if you can set up a one-to-one correspondance between the set you are counting and a set whose size you know. He let "Aleph-0" stand for the size of the set of counting numbers {1,2, etc} Using Cantor's method, if there are an Aleph-0 number of mailboxes, then there are an Aleph-0 number of items of mail. Proof. Assume mail boxes have been numbered 1,2,etc. number the 5 items of mail in each box by 5*(mail box#) - 4 5*(mail box#) - 3 5*(mail box#) - 2 5*(mail box#) - 1 5*(mail box#) now every positive integer is associated with some item of mail and vice versa. This is also true for mail boxes. Therefore the number of mail boxes equals the number of items of mail. Equivalently, you can prove it by removing each mail item #n from its box and placing it in mail box #n. Now each box has one item, so they must be the same. However ... that would take an infinite amount of time to actually do. At any given time, we only see a finite number of mail boxes and items, so I'm not really comfortable in saying you can really continue this process for ever! Have you heard of Hilbert's hotel? It had an Aleph-0 number of rooms. All were taken. A man arrives at the door wanting a room. Hilbert moves the occupant of room #1 into room #2 and the occupant of room #2 into room #3 etc. Now room #1 is free and the man gets it. A coach carrying an Aleph-0 number of people turn up, each wanting a room. Hilbert simply moves the occupant of room #1 to room #2, room #2 to room #4, room #3 to room #6 etc. All occupants move to an even numbered room. Now an Aleph-0 number of odd numbered rooms are free for the coach party. Suddenly an Aleph-0 number of coaches turn up, each containing an Aleph-0 number of people. You can't get me that way, says Hilbert. He moves occupant #n to rooom #((n*(n+1))/2) (#1 goes to #1, #2 goes to #3, #3 goes to #6 etc) Now there is one room free between #1 and #3 (gap #1) two rooms free between #3 and #6 etc (gap #2) He asks each i'th person on the j'th bus to move into the ith room free in the (i+j-1)th gap All now have a room again. Hilbert has finally had enough and puts up a sign. "No vacancies!" After all, after this he has an Aleph-0 number of dollars and can now retire for the rest of eternity!

Suppose each mailbox had 5 items in it. Remove all the items from the mailboxes and then place one item in each mailbox. You would have no leftover items and no empty mailboxes. Would that mean the number of items is the same as the number of mailboxes?

yes