We can prove that a certain language is non regular by using a theorem called “Pumping Lemma”. According to this theorem every regular language must have a special property. If a language does not have this property, than it is guaranteed to be not regular. The idea behind this theorem is that whenever a FA process a long string (longer than the number of states) and accepts, there must be at least one state that is repeated, and the copy of the sub string of the input string between the two occurrences of that repeated state can be repeated any number of times with the resulting string remaining in the language.
Pumping Lemma :
Let L be a regular language. Then the following property olds for L.
There exists a number 
(called, the pumping length), where, if
w is any string in L of length at least k i.e. 
, then w may be divided into three sub strings w = xyz, satisfying the following conditions:
i.e. 
