  The Reliable Broadcast concept allows an honest party to send a message to
all other parties and to make sure that all honest parties receive this
message. In addition, it allows an honest party that received a message to know
that all other honest parties would also receive the same message. This
technique is important to ensure distributed consistency when facing failures.
  In the current paper, we study the ability to use \RR to consistently
transmit a sequence of input values in an asynchronous environment with a
designated sender. The task can be easily achieved using counters, but cannot
be achieved with a bounded memory facing failures. We weaken the problem and
ask whether the receivers can at least share a common suffix. We prove that in
a standard (lossless) asynchronous system no bounded memory protocol can
guarantee a common suffix at all receivers for every input sequence if a single
party might crash.
  We further study the problem facing transient faults and prove that when
limiting the problem to transmitting a stream of a single value being sent
repeatedly we show a bounded memory self-stabilizing protocol that can ensure a
common suffix even in the presence of transient faults and an arbitrary number
of crash faults. We further prove that this last problem is not solvable in the
presence of a single Byzantine fault. Thus, this problem {\bf separates}
Byzantine behavior from crash faults in an asynchronous environment.
