Prove that $w_{2n}(\gamma^n\oplus \gamma^n)\neq 0$

Question

Let $\gamma^n$ be the canonical $n$-plane bundle over the infinite Grassmann manifold $G_n(\mathbb{R}^{\infty})$. I'm asked to prove that

$$w_{2n}(\gamma^n\oplus \gamma^n)\neq 0$$

(exercise 9-A from Milnor-Stasheff)

I started writing down a proof: $w_{2n}(\gamma^n \oplus \gamma^n)=\sum_{i+j=2n} w_i(\gamma^n)\smile w_j(\gamma^n)$, but being $w_k(\gamma^n)=0$ for $k> n$, we have $$w_{2n}(\gamma^n \oplus \gamma^n)=w_n(\gamma^n)\smile w_n(\gamma^n)$$ How do I prove that this is not zero? it's enough to say that $w_n(\gamma^n)^2\neq 0 \in H^*(G_n(\mathbb{R}^{\infty})$ (because it's the polynomial algebra generated by the Stiefel-Whitney class up to degree $n$)?

It seems a little bit sloppy as an argument, but I cannot think anything else right now

Answer 1

In order to mark this question as answered, the above reasoning is correct. (Thanks to Qiaochu Yuan for the confirm)