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

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