sequences of eventually monotone functions
A sequence is eventually monotone if there exists an N such that (a_n) is monotone for n greater than N.
Let (X,M) be a measurable space and let fn be a sequence of measurable functions on X. Show that the set of points where fn is eventually monotone is a measurable set.
Here is my idea: view only the set of points on which fn is eventually monotone. Then pointwise fn converges to some f on this set. Can I conclude the measurability of f and hence the set of points where fn is eventually monotone from this?