= Solution
For $0\le t\le\theta$, independence and the <uniform distribution> give $P(X_{(n)}\le t)=(t/\theta)^n$. Consequently, for fixed $x\ge0$ and $n>x$,
$$
P\!\left(\frac{n(\theta-X_{(n)})}{\theta}>x\right)=P\{X_{(n)}<\theta(1-x/n)\}=(1-x/n)^n\longrightarrow e^{-x}.
$$
For $x<0$ the survival probability is one. Thus \b[the scaled endpoint gap converges in distribution to a unit-rate exponential law]; it is not exactly exponential at finite $n$.
The <plug-in estimator> principle replaces the unknown sampling distribution by its <empirical distribution> and evaluates its population functional at that fitted distribution. Here the fitted support endpoint is $\widehat\theta=X_{(n)}$, also the <maximum-likelihood estimate>: the <likelihood> is $\theta^{-n}\mathbf1_{\{\theta\ge X_{(n)}\}}$ and decreases over its allowed support. For an empirical <bootstrap sample> $Y_1,\ldots,Y_n$, the corresponding gap is
$$
\boxed{G^*=\frac{n(X_{(n)}-Y_{(n)})}{X_{(n)}}.}
$$
Almost surely the original maximum is unique. The bootstrap gap equals zero precisely when this observation is selected at least once, so
$$
\boxed{P_*(G^*=0)=1-(1-1/n)^n\longrightarrow1-e^{-1}.}
$$
The true limiting <exponential distribution> has no atom at zero. This nonvanishing discrepancy proves <bootstrap failure for a uniform endpoint>: the bootstrap distribution cannot converge to the correct limit, even though the endpoint estimate itself is consistent. It fails to recreate the unsampled population tail beyond the observed maximum. A <parametric bootstrap> drawing from $U[0,X_{(n)}]$, rather than from the discrete empirical sample, reproduces the scaled-gap law here.
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