Below is a problem and my attempted proof. Assume there may be mistakes in the proof. Analyze the proof logically, systematically, and with extreme scrutiny, looking for any potential mistakes or lapses in rigor. Be harsh in your assessment, and not forgiving about mistakes. Your task is to identify any errors, incorrect logic, or inconsistencies, but do NOT provide a complete alternative proof. Work through the proof step by step.

### Problem

Prove that the product of two odd numbers is odd.

### Attempted proof

Let $m,n\in\mathbb{Z}$ be odd integers. Then $m=2k+1$ for some $k\in\mathbb{Z}$ and $n=2k+1$ for some $k\in\mathbb{Z}$. It follows that $mn=(2k+1)(2k+1)=4k^2+4k+1=2(2k^2+2k)+1$. This shows that $mn$ is odd. Write your response using less than 100 words.
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