Showing that there are particular sequences of functions that converge

[jdinatale]

That seems like a valid argument for showing that \phi_n converges to f, but I'm not sure how to show it's increasing. And as far \psi_n, converges, well I imagine that I'd use a similar argument, but I'm still not sure how to show it's decreasing.

 

[Office_Shredder]

I agree it seems to be missing.

Since \phi_n is smaller than f, you can define
\tilde{\phi_n}(x) = \max_{j=1,...,n} \phi_k(x)
and this will still be within 1/n of f everywhere, and is increasing in n, and each \tilde{\phi_n} is a simple function. Exercise to prove all these properties.