Singular Values/l2-norm of Pseudo-inverse

I am trying to prove, given a matrix $A=\lbrack\frac{A_1}{A_2}\rbrack\in C^{m\times n}$, with $A_1\in C^{n\times n}$ non-singular, that:

$||A^+||_2\leq||A_1^{-1}||_2$

($||\cdot||_2$ is the induced $\ell_2$ norm, $(\cdot)^+$ is the Moore-Penrose pseudoinverse.)

Supposed to be simple but I’m having trouble relating $A$’s singular values to $A_1$’s. Any ideas?

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