For two random variables $X$ and $Y$
We know that $\rho = \frac{Cov(X,Y)}{\sigma_X \sigma_Y} = 1 $ if and only if $X = cY$ for some constant $c$.
If $X = cY$, then $\rho = 1$ is easy to prove.
If $\rho = 1$, then how do we prove $X = cY$ for some constant c?
EDIT: Can we even prove that?
If $\rho = 1$, then at best we can prove that $X - E[X] = c (Y - E[Y])$ for some constant c?