Pseudoinverse of a matrix: Prove existance of B ∈ R^(Nxr) and C ∈ R^(rxp)

[Moresco]

Suppose A is an Nxp matrix of rank r with real coefficients. The matrix A# is called a pseudoinverse of A.

Prove the existence of the matrices

B ∈ R^Nxr and C ∈ R^rxp for which:​

A = BC
det(CC^T) != 0
det(B^TB) !=0

Find the pseudoinverses of B and C. Also show that A# = C#B# is a pseudoinverse for A.

I'm trying this with the SVD of A, but I'm not sure how to represent B & C in the context of the SVD and carry on from there.