Necessary and sufficient memory size for m-hypothesis testing

Let
X_1,X_2, \cdots
be a sequence of independent identically distributed Bernoulli random variables with Pr
\{X_i = 1 \} = p
. Consider the
m
-hypothesis test
H_1 : p < p_1, H_2 : p_1 < p < p_ 2, \cdots
versus
H_m: p_{m-1} < p < p_m
. It is shown that, for a time-varying finite memory,
m + 1
states are necessary and sufficient to resolve the correct hypothesis with a zero-limiting probability of error.