Let $\underline{a}_N:=(a_1,...,a_N)^\top \in \mathbb{R}^N$ denote a vector of lenght $N$.
Define $ x_N^* := \arg \min_{x \in \mathbb{R}} \left\| \underline{a}_N - x \mathbb{1}_N \right\|_p $
where $\mathbb{1}_N = (1,1,...,1)^\top \in \mathbb{R}^N$ and $\left\| \cdot \right\|_p$ is the $p$-norm, defined as $\left\| y \right\|_p = \left( \sum_{i=1}^N |y|^p \right)^{1/p}$.
Also define $ \xi_N^* := \arg \min_{x \in \mathbb{R}} \left\{\left\| \underline{a}_N - x \mathbb{1}_N \right\|_p + x \right\}$
Prove that $ \lim_{N \rightarrow \infty} |x_N^* - \xi_N^*| = 0$
Notes: I tried differentiating $\left\| \underline{a}_N - x \mathbb{1}_N \right\|_p$ with respect to $x$, but then you get $|a_i - x|$ that is not differentiable. I mean that if we assume that $a_i - x > 0 \ \forall i$ then we don't need the absolute value in the norm definition.