Let a hotel have a denumerable set of rooms numbered 1, 2, 3, .... Then any finite number of guests can be accommodated without evicting the current
guests by moving the current guests from room to room . Furthermore, a denumerable
number of guests can be similarly accommodated by moving the existing guests from
to , freeing up the denumerable
number of rooms .
Even countably many buses, each carrying countably many guests, can be accommodated. Numbering the buses and their seats by positive integers and , respectively, the guest in seat of bus can be assigned to room after the original guests are moved to the odd-numbered
rooms. This is an explicit illustration that a countable union of countable
sets is countable. By contrast, guests indexed by all infinite sequences of 0s
and 1s cannot be accommodated, since those sequences form an uncountably
infinite collection.
of guests can be accommodated without evicting the current
guests by moving the current guests from room 
to room 
. Furthermore, a denumerable
number of guests can be similarly accommodated by moving the existing guests from
to 
, freeing up the denumerable
number of rooms 
.
and 
, respectively, the guest in seat 
of bus 
can be assigned to room 
after the original guests are moved to the odd-numbered
rooms. This is an explicit illustration that a countable union of countable
sets is countable. By contrast, guests indexed by all infinite sequences of 0s
and 1s cannot be accommodated, since those sequences form an uncountably
infinite collection.
