RKHS of limit of reproducing kernel

Question

Let $\{K_n\}$ be a sequence of positive definite kernels such that

1. $K_n\preccurlyeq K_{n+1}$ for all $n$ (it means that $K_{n+1}-K_n$ is a positive definite kernel)
2. They have a limit $K(x,y)=\lim_{n\to \infty}K_n(x,y)$.

If $K$ is a positive definite kernel, we write $\mathcal{H}_K$ for its associated reproducing kernel Hilbert space (RKHS).

My question is: is the following equation true $$\mathcal{H}_K = \bigcup_{n=1}^\infty \mathcal{H}_{K_{n}}\,,$$ where $=$ means equality as a set.

And what about decreasing $\{K_n\}$? In other words, $K_n\succcurlyeq K_{n+1}$ and they have a limit $K$. Is it true that $$\mathcal{H}_K=\bigcap_{n=1}^\infty \mathcal{H}_{K_n}\,.$$