Compact space which is not sequentially compact

Greets

So this is exercise 17G.1 of Stephen Willard's "general topology", and it's stated:

Show that there is a compact space that is not sequentially compact

[Hint: Consider an uncountable product of copies of $[0,1]$]

So here is my answer

Let $S$ be the set of all strict increasing sequences of natural numbers, and for each $s\in{S}$ let $X_s=\left\{{0,1}\right\}$ with the discrete topology, put $X=\prod_{s\in{S}}X_s$. Then $X$ is compact by Tychonoff's theorem. Let us see that $X$ is not sequentially compact. Define $\left\{{x_n}\right\}_{n\in{\mathbb{N}}}$ as follows: let $s\in{S}$ with $s=\left\{{n_k}\right\}_{n\in{\mathbb{N}}}$, then define $(x_n)_s=0$ if $n=n_k$ for some $k$ even and define $(x_n)_s=1$ otherwise. Let us see that $\left\{{x_n}\right\}_{n\in{\mathbb{N}}}$ has no convergent subsequence in $X$. Let $s\in{S}$ with $s=\left\{{n_k}\right\}_{n\in{\mathbb{N}}}$, then $(x_{n_k})_s=0$ for $k$ even and $(x_{n_k})_s=1$ for $k$ odd, thus $\left\{{x_{n_k}}\right\}_{k\in{\mathbb{N}}}$ does not converge in $X=\prod_{s\in{S}}X_s$ since it doesn't converge componentwise.

As you can see this example is too straightforward, and I would like to see other examples.

Thanks.