Let \( A \) be an \( n \times n \) Hermitian matrix and \( \lambda_1 (A) \geq \lambda_2 (A) \geq \dots \geq \lambda_n (A) \) the eigenvalues of \( A \). Prove that for any \( 1 \leq k \leq n \)

\[

A \mapsto \lambda_1 (A) + \lambda_2 (A) + \dots + \lambda_k (A)

\]

is a convex function.

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