Let $(X, d)$ be a metric space and $\mathcal{M} :=\mathcal{M}(X)$ the space all non-negative finite Borel measures on $X$. The Prokhorov metric $d_P$ on $\mathcal{M}$ is defined by $$ d_{P}(\mu, \nu) := \inf \left\{ \alpha>0 \,\middle\vert\, \begin{align*} \mu(A) \leq \nu\left(A_{\alpha}\right)+\alpha \\ \nu(A) \leq \mu \left(A_{\alpha}\right)+\alpha \end{align*} \quad \forall A \in \mathcal{B}(X) \right\}, $$ with $A_{\alpha} := \{x \mid d(x, A)<\alpha\}$ and $d(x, A) := \inf \{d(x, a) \mid a \in A\}$.
Previously, I proved that $d_P$ is indeed a metric. In my lecture note, $d_P$ has been computed for Dirac probability measures, i.e., $$ d_P (\delta_x, \delta_y) = \min\{d(x, y), 1\} \quad \forall x,y \in X. $$
Now I try to compute it for finite Dirac measures, i.e.,
Let $\mu := a \delta_x$ and $\nu := b \delta_y$ with $a,b \ge 0$ and $x,y \in X$. Compute $d_P(\mu, \nu)$.
Could you have a check on my attempt?
I post my proof separately as below answer. If other people post an answer, of course I will happily accept theirs. Otherwise, this allows me to subsequently remove this question from unanswered list.