I am trying to follow a proof of this result (i.e. for $f,g$ in an inner product space, $|\langle f,g\rangle|\le \|f\|\cdot\|g\|$) in the text "Theory of Linear Operators in Hilbert Space" by Akhiezer and Glazman. The part I am having trouble with is (paraphrased for brevity),
"We find that for any real $\lambda$, $0\le \lambda^2\langle g,g\rangle + 2\lambda|\langle f,g\rangle| + \langle f,f\rangle$, which implies that $|\langle f,g\rangle|^2 \leqslant \langle f,f\rangle\cdot\langle g,g\rangle$."
I am assuming there is a clever choice of $\lambda$ which yields the desired result but I am just not seeing it.
Hint: Think of $\lambda^2\langle g,g\rangle + 2\lambda|\langle f,g\rangle| + \langle f,f\rangle$ as a quadratic polynomial in $\lambda$ and try to solve it.