Matrix multiplication and nilpotent matrix

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Given that you have an $n\times n$-matrix called $A$ with the property that $A^k = 0$. Prove that $A^n = 0$.

Have heard about eigenvalues and null space as ways for the proof. The thing is that I can't wrap my head around how to use null space and we haven't learned about eigenvalues yet. I appreciate all the help I can get.