= Solution
{c}
A family $(\xi_i)_{i\in I}$ of <integrable random variables> is <uniformly integrable> when
$$
\boxed{\lim_{K\to\infty}\sup_{i\in I}\mathbb E\bigl[|\xi_i|\mathbf1_{\{|\xi_i|>K\}}\bigr]=0.}
$$
The <Lipschitz condition> gives, on $\{X<1\}$,
$$
|Z_n|=2^n|f(X_n+2^{-n})-f(X_n)|\leq2^nL2^{-n}=L.
$$
Our chosen zero value on $\{X=1\}$ satisfies the same bound. Hence the <random variables> $(Z_n)$ are <integrable>, and for $K\geq L$ every tail <expectation> in the definition vanishes. Therefore \b[the family is uniformly integrable].
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