= Solution
Split the overlapping products into two nonoverlapping families:
$$
U_n=\sum_{j=1}^nX_{2j-1}X_{2j}+\sum_{j=1}^nX_{2j}X_{2j+1}.
$$
Within each sum the summands depend on disjoint independent pairs and are independent identically distributed Bernoulli variables of mean $1/4$. The two families need not be independent of each other. The <strong law of large numbers> says that averages of independent identically distributed integrable random variables converge <almost surely> to their <expectation>. Applying it separately to each family and intersecting the two probability-one events gives
$$
\boxed{\frac{U_n}{n}\longrightarrow\frac14+\frac14=\frac12\quad\text{almost surely}.}
$$
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