Answer:
This statement can be proven by contradiction for  (including the case where
 (including the case where  .)
.)
 .
.
 :
:
 .
.
 .
.
 .
.
 .
.
 .
.
 .
.
 .
.
 .
.
 .
.
 .
.
 .
.
 .
.
 .
.
 .
.
 .
.
 .
.
 .
.
Step-by-step explanation:
Assume that the natural number  is a perfect square. Then, (by the definition of perfect squares) there should exist a natural number
 is a perfect square. Then, (by the definition of perfect squares) there should exist a natural number  (
 ( ) such that
) such that  .
.
Assume by contradiction that  is indeed a perfect square. Then there should exist another natural number
 is indeed a perfect square. Then there should exist another natural number  such that
 such that  .
.
Note, that since  ,
,  . Since
. Since  while
 while  , one can conclude that
, one can conclude that  .
. 
Keep in mind that both  and
 and  are natural numbers. The minimum separation between two natural numbers is
 are natural numbers. The minimum separation between two natural numbers is  . In other words, if
. In other words, if  , then it must be true that
, then it must be true that  .
.
Take the square of both sides, and the inequality should still be true. (To do so, start by multiplying both sides by  and use the fact that
 and use the fact that  to make the left-hand side
 to make the left-hand side  .)
.)
 .
.
Expand the right-hand side using the binomial theorem:
 .
.
 .
.
However, recall that it was assumed that  and
 and  . Therefore,
. Therefore, 
 .
.
 .
.
Subtract  from both sides of the inequality:
 from both sides of the inequality:
 .
.
 .
.
Recall that  was assumed to be a natural number. In other words,
 was assumed to be a natural number. In other words,  and
 and  must be an integer. Hence, the only possible value of
 must be an integer. Hence, the only possible value of  would be
 would be  .
.
Since  could be equal
 could be equal  , there's not yet a valid contradiction. To produce the contradiction and complete the proof, it would be necessary to show that
, there's not yet a valid contradiction. To produce the contradiction and complete the proof, it would be necessary to show that  just won't work as in the assumption.
 just won't work as in the assumption. 
If indeed  , then
, then  .
.  , which isn't a perfect square. That contradicts the assumption that if
, which isn't a perfect square. That contradicts the assumption that if  is a perfect square,
 is a perfect square,  would be a perfect square. Hence, by contradiction, one can conclude that
 would be a perfect square. Hence, by contradiction, one can conclude that 
 .
.
Note that to produce a more well-rounded proof, it would likely be helpful to go back to the beginning of the proof, and show that  . Then one can assume without loss of generality that
. Then one can assume without loss of generality that  . In that case, the fact that
. In that case, the fact that  is good enough to count as a contradiction.
 is good enough to count as a contradiction.