How to prove inverse direction for correlation coefficient?

Question

To show: If |Cor(X,Y)| = 1, then there exists a, b ∈ R s.t Y = bX + a. Any ideas or hints to proceed?

Basically, I've to prove that if the absolute value of correlation b/w two random variables is 1, then they should be linearly related.

So far,

$$ |cor(X, Y)| = 1 $$

$$ \frac{|Cov(X, Y)|}{|\sigma(X) \sigma(Y)|} = 1 $$

$$ |Cov(X, Y)| = |\sigma(X) \sigma(Y)| $$

How to proceed further?