I have a Ridge regression model to estimate the coefficients of the true model $y = X\beta + \epsilon$. I have the standard model where $\mathbb{E}[\epsilon] = 0, \ \mathrm{Var}(\epsilon) = I.$ The ridge estimator of $\beta$ is: $\beta^\mathrm{Ridge} = (X^\top X + \lambda I )^{-1} X^\top y$
Assume we have a fixed testing point $x_0$. I have proved that by increasing $\lambda$ the variance of estimation $$\hat{f}(x_0) = x_0^\top (X^\top X + \lambda I)^{-1} X^\top y$$ is decreasing.
Now I want to show that by increasing $\lambda$ the squared bias of the test estimation steadily increase.
I thought of using the bias-variance tradeoff, but it does not work since the tradeoff tells us $$Error(x_0) = \text{Irreducible Error} + \mathrm{Bias}^2(\hat{f}(x_0)) +\mathrm{Variance}(\hat{f}(x_0)) . $$ To show that increased variance implies decreased bias, we need to have the same $Error(x_0)$ but this is not the case.
So, how can I show that the bias of our ridge estimation on the test data steadily increases with increasing $\lambda$?