Measurability Question?

I assigned the following to a class I'm teaching and, to my embarrassment, I cannot come up with a solution.

Let $(X,\mathcal B)$ be a measurable space and let $(f_n)_{n\ge 1}$ be a sequence of measurable functions from $X$ to $\mathbb R$. Let $S=\{x\colon \text{$(f_n(x))$ contains a (strictly) increasing subsequence}\}$. Is $S$ necessarily measurable?

I believe the answer is "no", but I find myself ill-equipped to prove that a set is not measurable. Any suggestions?

The set $A = \{x \in \mathbb{Q}^{\omega} : \{ x(n) : n < \omega \} \text{ is reverse well order as a suborder of rationals} \}$ is not Borel. Luzin and Sierpinski showed that it is in fact $\Pi_1^1$-complete - see Kechris, Classical descriptive set theory, page 213. Now let $f_n:\mathbb{Q}^{\omega} \to \mathbb{R}$ be $f_n(x) = x(n)$.