= Solution
The <polynomial> $p_n$ interpolates $f$ at the <interpolation nodes> when
$$
p_n(x_i)=f(x_i)\qquad(0\leq i\leq n).
$$
For distinct <interpolation nodes>, define the <Lagrange cardinal polynomials>
$$
\ell_i(x)=\prod_{j\ne i}\frac{x-x_j}{x_i-x_j}.
$$
The denominators are nonzero, each <polynomial degree> is at most $n$, and $\ell_i(x_j)=\delta_{ij}$, with $\delta_{ij}$ the <Kronecker delta>. Therefore
$$
\boxed{p_n(x)=\sum_{i=0}^nf(x_i)\ell_i(x)}
$$
is a <Lagrange interpolation polynomial> belonging to the <vector space of univariate polynomials> $\Pi_n$, and it has all the required sampled values.
If $r_n\in\Pi_n$ also works, $p_n-r_n$ has $n+1$ distinct <roots of a polynomial> while its <polynomial degree> is at most $n$. The <root bound for a polynomial> forces it to be the <zero polynomial>, proving uniqueness. The phrase “degree $n$” denotes the allowed maximum; the actual <polynomial degree> may be smaller. No <continuity> assumption is needed merely to construct the <interpolation polynomial> from these finitely many values.
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