pure maths:prove√10 is irratio

Suppose √10 is rational and can be expressed as a quotient of 2 integers = a/b where a and b are integers and relative prime, i.e. a/b is the most reduced form of the quotient.
√10 = a/b => b√10 = a
Squaring both sides,
10b^2 = a^2, is even
Therefore a can be expressed as a = 2c where c is an integer
10b^2 = (2c)^2
10b^2 = 4c^2
5b^2 = 2c^2 is also even
Since 5 is odd, b^2 is even
Therefore b is also even and can be expressed as b = 2d where d is an integer.
It follows that √10 = a/b = 2c/2d which is a contradiction that a and b are relative prime.
Therefore √10 is not rational