lim of simple functions converge to nonnegative measurable function

In an earlier post I mentioned this result

Let f be a nonnegative function, then there is an increasing sequence
{phi_n} of nonnegative simple functions each of which vanishes outside
a set of finite measure such that f = lim phi_n.

This appears to be quite easy if we drop the requirement that {phi_n}
has to be an increasing sequence. It seems that there should be an
explicit way to construct these simple functions, however all of my
attempts have failed to be increasing.

What is the secret behind this?

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