Measurable Spaces – Problem (44/365)

Following on from the previous post, is the following a Borel set?

$\displaystyle \{ x \in R^\infty : \lim_n x_n > a \}$

This is a Borel set because we can intersect the set of converging sequences and the set of sequences bounded from below.

$\displaystyle \{ x \in R^\infty : \lim_n x_n > a \} = \{ x \in R^\infty : x_n \rightarrow \} \cap \{ x \in R^\infty : \sup_n \inf_{k \ge n} x_k > a \}$