Consider a sequence $\{ f_k \}_{k=1}^{\infty}$ of locally-bounded functions $f_k: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$.
Assume the following.
For any sequence $\{X_k\}_{k=1}^{\infty}$ of compact sets $X_k \subset \mathbb{R}^n$ such that $X_k \subseteq X_{k+1}$ and $X_k \rightarrow \mathbb{R}^n$, there exist (a uniform) $M \in \mathbb{R}_{>0}$ such that
$ \sup_{x \in X_k} f_k(x) \leq M$
Say if the following claim holds (or find a counterexample).
There exists $K \in \mathbb{Z}_{\geq 1}$ such that
$ \sup_{x \in \mathbb{R}^n} f_K(x) < \infty$
Note: can we use this argument?