proof of limits involving infinity using formal definition

I am reading my lecture notes on limits and stumbled across a 'solved' example problem.

It concerns with the limit of $$ \lim_{x \to \infty} (x - \sqrt{x + 1}) = \infty$$

Using the formal definition of $ \lim_{x \to \infty} f(x) = \infty $, as $ \forall A > 0, \exists M > 0 \textrm{ such that } x > M \implies f(x) > A $

My lecturer has given $M = A^2$ as a possible M, without justification.

I am having difficulty in finding this, from considering $$ x - \sqrt{x+1} = \frac{x^2 - x - 1}{x+\sqrt{x+1}} $$

Thanks for any ideas.