I am wondering whether the following sequence converges in $D([0,\infty), \mathbb{R})$ with J1 topology:
$ x_n:=\frac12\mathbb{1}_{[1,1+1/n)}+\mathbb{1}_{[1+1/n,\infty)} $
I would expect the limit to be $\mathbb{1}_{[1,\infty)}$, but I am having problems in proving compactedness of the sequence $\{x_n\}_{n\in\mathbb{N}}$ since the supremum over $\{x_n\}_{n\in\mathbb{N}}$ of the modulus of continuity does not seem to vanish when $\delta$ goes to zero (see for example "Convergence of probability measures", P. Billingsley (1999), Theorem 12.3).
Is this a problem due to multiple jumps collapsing in the limit or am I simply missing something?
This sequence indeed does not converge in the $J_1$ topology. To see this, note that for any increasing homeomorphism $\lambda:[0,\infty)\rightarrow[0,\infty)$ there must exist some $t$ such that $x_n(\lambda(t))=\frac12$. But since the supposed limit $x=\mathbb 1_{[1,\infty)}$ never takes on the value $\frac12$, we must have $$\Vert x_n\circ\lambda -x\Vert_\infty\geq \frac12$$ and therefore it cannot hold that $x_n\to x$ in $J_1$.
It is indeed not possible for jumps to "collapse" in the $J_1$ topology. If I remember correctly, this is why one may want to introduce the $J_2$-topology, where such a collapse of jumps is allowed. Though your specific example may also not converge in $J_2$ (I am not entirely sure).
For an, in my opinion, excellent introduction to (some of) the Skorokhod topologies, see: "The Skorokhod Topologies: What They Are and Why We Should Care"