How to show that the Gaussian Process parameters decreases with more training data

Question

The posterior mean prediction of a Gaussian Process is given by

$$\mu(x_*) = \sum_{i=1}^n\alpha_i k(\mathbf{x}_i,\mathbf{x}_*) $$ where $$\alpha = (K + \sigma_n^2I)^{-1} \mathbf{y}$$

Can we show that $\|\alpha \| $ decreases with more training data? i.e. $$\| \alpha_{n + 1} \| < \| \alpha_{n} \|$$