= Solution
Let $\{|u_j\rangle\}_{j=1}^N$ be another complete <orthonormal basis>, and write $|u_j\rangle=\sum_n U_{nj}|n\rangle$. Orthonormality makes $U$ a <unitary matrix>, so $UU^\dagger=I$. Therefore
$$
\begin{aligned}
\sum_j|u_j\rangle\langle u_j|
&=\sum_{n,k}\left(\sum_jU_{nj}U_{kj}^*\right)|n\rangle\langle k|\\
&=\sum_n|n\rangle\langle n|=I_N.
\end{aligned}
$$
Thus \b[the resolution of the identity is independent of the orthonormal basis]. Equivalently, expanding any vector in the new complete basis shows that the sum of its <orthogonal projections> returns that vector.
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