Prove that the decimal expansion of a rational number 
eventually is repeating.
Proof:
Write
, where 
is the integer part of, and 
is the decimal part of .
Let
, 
, 
, …, 
, …where 
for any 
is
the set of remainders for integers divided by 
.
Since there are (including 0) possible remainders
for integers divided by, there are at least two remainders from set B are the same
by Pigeonhole Principle.
Let the two remainders are 
and 
with
. Without lose of
generality, assume that 
.
and 
. Since
, we have 
and 
by Division
principle. The same argument, 
and 
,…, 
and 
, 
and 
, … The decimal expansion is repeating.