Convergence of jump function with respect to the Skorokhod metric

Question

Consider the space $D := D([0, 1], \mathbb{N})$ of cadlag functions $f : [0, 1] \to \mathbb{N}$ equipped with the Skorokhod $J_1$-metric $d(f,g) := \inf_{\lambda \in \Lambda} \max \{ \lVert \lambda - id \rVert, \lVert f \circ \lambda - g \rVert \}$ where $\Lambda$ is the set of continuous strictly increasing functions $\lambda : [0,1] \to [0,1]$ with $\lambda(0) = 0$ and $\lambda(1) = 1$ and $\lVert \cdot \rVert$ the supremum norm. Let $x_n, x \in D$ with $x_n \to x$ in this metric. Denote by $\tau_n$ and $\tau$ the timepoint of the first jump of $x_n$ and $x$ respectively. Is it true that $\tau_n \to \tau$ and $x_n(\tau_n) \to x(\tau)$?

Answer 1

Lemma: If $x_n \to x$ in $J_1$ topology and $x$ is not constant, then $\tau_n \to \tau$ and $x_n(\tau_n) \to x(\tau)$.

Proof: Since $x_n \to x$ in $J_1$ we can find a sequence $(\lambda_n)_{n \in \mathbb{N}}$ of continuous strictly increasing functions $\lambda_n: [0,1] \to [0,1]$ with $\lambda_n(0)=0$, $\lambda_n(1)=1$ such that $$\|\lambda_n-\text{id}\|_{\infty} \to 0 \qquad \|x_n \circ \lambda_n-x\|_{\infty} \to 0.$$ In particular, there exists $N \in \mathbb{N}$ such that $$\|x_n \circ \lambda_n-x\|_{\infty} < 1 \quad \text{for all $n \geq N$},$$ and since the functions are taking values in $\mathbb{N}$, this already implies $$ \forall n \geq N \, \forall t \in [0,1]: \quad x_n(\lambda_n(t))= x(t). \tag{1}$$ As $x$ is right-continuous, non-constant and takes values in $\mathbb{N}$, we know that the first jump time $\tau$ of $x$ satisfies $$\tau = \inf\{t \in (0,1]: x(0) \neq x(t)\} \in (0,1].$$ It follows from $(1)$ that $$\tau_n = \lambda_n(\tau)$$ and $$x_n(\tau_n) = x_n(\lambda_n(\tau)) = x(\tau)$$ for all $n \geq N$. In particular, $x_n(\lambda_n(\tau)) \to x(\tau)$ and $$\tau_n =\lambda_n(\tau) \xrightarrow[]{n \to \infty} \tau.$$