Do you know the definition of as the completion of under taking limits of Cauchy sequences?
If, so, just show that there exists a Cauchy sequence of rationals whose limit has a square equal to 2.
Let as e.g. Also is bounded above. exists. Note that since
Suppose Then and so such that Then
so contradicting the leastness of
Now suppose Choose a natural number such that and Then
so contradicting the fact that is an upper bound for
Since both and lead to a contradiction, we conclude that