= Solution
The <empirical spectral measures> of the two real <symmetric matrices> are
$$
L_N=\frac1N\sum_{i=1}^N\delta_{\lambda_i},\qquad \widehat L_N=\frac1N\sum_{i=1}^N\delta_{\widehat\lambda_i},
$$
where $\delta_t$ is the <Dirac measure> at $t$. Thus $\langle L_N,f\rangle=N^{-1}\sum_i f(\lambda_i)$. Pair the <eigenvalues> in the specified increasing order and apply the <triangle inequality> followed by <Lipschitz continuity> with bound $1$:
$$
\boxed{\left|\langle L_N,f\rangle-\langle\widehat L_N,f\rangle\right|\leq\frac1N\sum_{i=1}^N|f(\lambda_i)-f(\widehat\lambda_i)|\leq\frac1N\sum_{i=1}^N|\lambda_i-\widehat\lambda_i|.}
$$
This deterministic estimate uses only the <Lipschitz bound>; no entry independence or distributional hypothesis is needed here.
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