Assume we have a non-decreasing net of semicontinuous functions $f_\alpha:[0,1]\to\mathbb{R}$ such that $\lim_\alpha f_\alpha\to f$ pointwise. Please is it true that one can extract a countable sequence $\{f_n\}\subset \{f_\alpha\}$ such that $f_n\to f$ ? Any reference?

Assume we have a non-decreasing net of lower semicontinuous functions $f_\alpha:[0,1]\to\mathbb{R}$ such that $\lim_\alpha f_\alpha\to f$ pointwise. Please is it true that one can extract a countable sequence $\{f_n\}\subset \{f_\alpha\}$ such that $f_n\to f$ ? Any reference?