real analysis - the conditions for a measurable function to be the uniform limit of simple functions

In our homework we are asked to prove that, on a measurable space $(\Omega,\mathcal{F})$, every function $f:\Omega \rightarrow R, f\geq 0$ can be written as the uniform limit of an increasing limit of simple functions.

However I looked up $Real\ Analysis$ by Royden and baby Rudin, and here a boundedness condition is needed. Is it true? or the conclusion holds even without the boundedness condition?