Recently I took a complex analysis exam, and one of the problems was to prove that $$\frac{1}{\Gamma(s)} = s \prod_{n = 1}^\infty \frac{\left(1 + \frac{s}{n}\right)}{\left(1+ \frac{1}{n}\right)^s}$$ I was allowed to use the Stein-Shakarchi text, in which chapter 6, theorem 1.7 states that for all $ s \in \mathbb{C}$, $$\frac{1}{\Gamma (s)} = e^{\gamma s}s \prod_{n = 1}^\infty \left(1 + \frac{s}{n}\right)e^{ - s/n},$$ where $\gamma$ is the Euler-Mascheroni constant. My professor took away points due to lack of rigor in one of the equalities, and said to ask other mathematicians what they thought about the proof. My proof is as follows:
We have that $$\frac{1}{\Gamma (s)} = e^{\gamma s}s \prod_{n = 1}^\infty \left(1 + \frac{s}{n}\right)e^{ - s/n} = e^{\lim_{N \to \infty} \sum_{k = 1}^N s/k - s\log N} s \prod_{n = 1}^\infty \left(1 + \frac{s}{n}\right) e^{- s/n}$$ The exponential terms cancel out as in the limit as $N \to \infty$ as we have that each term in the product is matched by a term from the Euler-Mascheroni term. Thus this equals $$\lim_{N \to \infty} e^{\log N^{-s}} s \prod_{n = 1}^\infty \left(1 + \frac{s}{n}\right) = \lim_{N \to \infty} s \prod_{n = 1}^\infty \frac{1}{N^s} \left(1 + \frac{s}{n}\right)$$ $N = N/(N - 1) \cdot (N - 1)/(N - 2) \cdots 3/2 \cdot 2/1 = \prod_{n = 1}^N (1 + 1/n)$, therefore as $N \to \infty$ we have that this equals $$s \prod_{n = 1}^\infty \frac{\left(1 + \frac{s}{n}\right)}{\left(1 + \frac{1}{n}\right)^s},$$ as desired.
My professor said that this equality $$\lim_{N \to \infty} e^{\log N^{-s}} s \prod_{n = 1}^\infty \left(1 + \frac{s}{n}\right) = \lim_{N \to \infty} s \prod_{n = 1}^\infty \frac{1}{N^s} \left(1 + \frac{s}{n}\right)$$ simply shows that $0 = 0$ and thus my argument is not rigorous. I was confused as to why this was because I only used standard rules involving limits and continuous functions, and it seems like the problems with the limit of $1/N^s$ approaching infinity are resolved in the proof of the equality that I started with (ch. 6, theorem 1/7) via the Hadamard factorization theorem. Is this rigorous or not? If so, what is my mistake? Thank you very much.