= Solution
The <sample maximum> has <cumulative distribution function>
$$
F_{M_n}(x)=\mathbb P(X_1\leq x,\ldots,X_n\leq x)
=\begin{cases}0,&x<0,\\x^n,&0\leq x\leq1,\\1,&x>1.\end{cases}
$$
The product $x^n$ follows from <independence of random variables> and the <continuous uniform distribution> of each observation. For every $x<1$, this <cumulative distribution function> tends to zero; for $x>1$ it equals one. These are exactly the values of the <cumulative distribution function> of the constant <random variable> $1$ at its continuity points. Therefore, by the definition of <convergence in distribution>,
$$
\boxed{M_n\xrightarrow{\mathrm d}1.}
$$
At $x=1$ the limiting <cumulative distribution function> is discontinuous, so the definition does not require checking that point, although $F_{M_n}(1)=1$ also holds.
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