Mathematicians define real numbers in an abstract way - as an 'ordered field' with 'the least upper bound property'. 
In physics, we use real numbers to represent distances. For us to be able to do this, we need to establish the following:

1. With every distance a real number can be associated.
2. With every real number, a distance can be associated.

In other words, a one to one correspondence. 
Is there any proof of this?