  Blockchain and other decentralized databases, known as distributed ledgers,
are designed to store information online where all trusted network members can
update the data with transparency. The dynamics of ledger's development can be
mathematically represented by a directed acyclic graph (DAG). One essential
property of a properly functioning shared ledger is that all network members
holding a copy of the ledger agree on a sequence of information added to the
ledger, which is referred to as consensus and is known to be related to a
structural property of DAG called one-endedness. In this paper, we consider a
model of distributed ledger with sequential stochastic arrivals that mimic
attachment rules from the IOTA cryptocurrency. We first prove that the number
of leaves in the random DAG is bounded by a constant infinitely often through
the identification of a suitable martingale, and then prove that a sequence of
specific events happens infinitely often. Combining those results we establish
that, as time goes to infinity, the IOTA DAG is almost surely one-ended.
