I want to show that
Let $M$ be a paracompact $T_2$ space and let $(K_i)_{i\in I}$ be a locally finite family of compact subsets of $M$, then there is a locally finite open covering $(U_j)_J$ of $M$ such that for every index $i\in I$ there is a $j\in J$ such that such that $K_i\subset U_j$.
Here is my approach:
For each $x\in K_i$, take an open neighbourhood $V_x$ of $x$ such that $|\{i\in I\mid K_i\cap U_x\neq\varnothing\}|<\infty$. Since $K_i$ is compact, there exists $x_1,\cdots, x_n$ such that $K_i\subset V_i=\bigcup_{i=1,\cdots,n}V_{x_i}$. Then $|\{i'\in I\mid K_{i'}\cap V_i\neq\varnothing\}|<\infty$. Since $(V_i)_I\bigcup\{M\backslash\bigcup_{i\in I}K_i\}$ is an open cover of $M$ and $M$ is paracompact, take a locally finite refinement $(W_j)_J\bigcup\{W'\}$ of that cover. Set $T_i=\{j\in J\mid K_i\cap W_j\neq\varnothing\}$ and $U_i=\bigcup_{j\in T_i}W_j$.
I want to that $(U_i)_I\cup\{W'\}$ is the locally finite cover we want, but I am stuck at showing the local finiteness.