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discrete math, which of these are contradiction proofs? I need help with them.

Proof 9.8: $⋆$ Prove $a_{n−k}=∑_{k}(ki )∇_{i}a_{n−1}$. Proof 9.9: $⋆$ Prove the following statement about the Catalan numbers $C_{n}$ for all all positive integers $n≥1:C_{n}≥2_{n−1}$ Proof 9.10: ᄎ Prove the following statement about the Catalan numbers $C_{n}$ for all all positive integers $n≥1:C_{n}≥n_{2}4_{n−1} $.

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