  Cake-cutting is a fundamental model of dividing a heterogeneous resource,
such as land, broadcast time, and advertisement space. In this study, we
consider the problem of dividing a discrete cake fairly in which the
indivisible goods are aligned on a path and agents are interested in receiving
a connected subset of items. We prove that a connected division of indivisible
items satisfying a discrete counterpart of envy-freeness, called envy-freeness
up to one good (EF1), always exists for any number of agents n with monotone
valuations. Our result settles an open question raised by Bil\`o et al. (2019),
who proved that an EF1 connected division always exists for the number of
agents at most 4. Moreover, the proof can be extended to show the following (1)
secretive and (2) extra versions: (1) for n agents with monotone valuations,
the path can be divided into n connected bundles such that an EF1 assignment of
the remaining bundles can be made to the other agents for any selection made by
the secretive agent; (2) for n+1 agents with monotone valuations, the path can
be divided into n connected bundles such that when any extra agent leaves, an
EF1 assignment of the bundles can be made to the remaining agents.
