= Solution
Take <independent and identically distributed random variables> $\xi_k$ with the <Bernoulli distribution> of parameter $1/2$, and their <natural filtration>. Set
$$
\boxed{M_n=2^n\mathbf1_{\{\xi_1=\cdots=\xi_n=1\}},\qquad M_0=1.}
$$
The event at $n=0$ is the whole sample space. <Independence> gives
$$
\mathbb E[M_{n+1}\mid\mathcal F_n]
=2^{n+1}\mathbf1_{\{\xi_1=\cdots=\xi_n=1\}}\,\frac12=M_n,
$$
so this is a nonnegative <martingale>; it is integrable with $\mathbb E M_n=1$. The first zero among the $\xi_k$ is almost surely finite, since the probability of having only ones through time $n$ is $2^{-n}\to0$. After that first zero, every $M_n$ is zero. Hence \b[the almost sure limit is zero]:
$$
\boxed{M_n\longrightarrow0\quad\text{almost surely},\qquad\mathbb E M_n=1.}
$$
This <fair-coin doubling martingale> also exhibits failure of <uniform integrability>: <convergence in L1> would force the <expectations> to converge to zero.
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