# Point-wise Convergence Timing

Consider a sequence of non-negative simply functions $$\{\phi_k(x)\}_{k=1}^\infty$$ that converges point-wise to a non-negative measurable function $$f$$.

$$\lim_{k\rightarrow\infty}\phi_k(x)=f(x)\space\space\forall x.$$

When we actually check this with a specific example, is the following timing of steps correct?

1. Fix $$x$$.
2. Let $$k$$ go to infinity and see if $$|\phi_k(x)-f(x)|<\varepsilon$$ for arbitrarily small $$\varepsilon$$.
3. Repeat step #1 and 2 for the entire $$x$$ in the domain of the function $$f$$.

Yes, if you remove step $$1$$ it checks the uniform convergence.