Let \( A, B, C = A+B \) be \( N \times N \) Hermitian matrices. Let \( \alpha_1 \geq \cdots \geq \alpha_N \), \( \beta_1 \geq \cdots \geq \beta_N \), \( \gamma_1 \geq \cdots \geq \gamma_N \) be the eigenvalues of \( A, B, C \), respectively. For any \( 1 \leq i, j \leq N \) with \( i+j -1 \leq N \), prove that
\[ \gamma_{i+j-1} \leq \alpha_i + \beta_j \]

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