Vtyjkyuk

Consider the space $D := D([0, 1], mathbbN)$ of cadlag functions $f : [0, 1] to mathbbN$ 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 $x_n(tau_n) to x(tau)$?

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 $(lambda_n)_n in mathbbN$ of continuous strictly increasing functions $lambda_n: [0,1] to [0,1]$ with $lambda_n(0)=0$, $lambda_n(1)=1$ such that $$|lambda_n-textid|_infty to 0 qquad |x_n circ lambda_n-x|_infty to 0.$$ In particular, there exists $N in mathbbN$ such that $$|x_n circ lambda_n-x|_infty < 1 quad textfor all $n geq N$,$$ and since the functions are taking values in $mathbbN$, this already implies $$ forall n geq N , forall t in [0,1]: quad x_n(lambda_n(t))= x(t). tag1$$ As $x$ is right-continuous, non-constant and takes values in $mathbbN$, we know that the first jump time $tau$ of $x$ satisfies $$tau = inft in (0,1]: x(0) neq x(t) in (0,1].$$ 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 $$tau_n =lambda_n(tau) xrightarrown to infty tau.$$