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Unexplained conclusion regarding an inequality

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Below we have the direct implication of Lebesque's Integrability Criterion, taken from: https://www.math.mcgill.ca/labute/courses/255w03/L10.pdf


How come does the author deduce that from:$$\frac{\epsilon}{2^i}\sum\limits_{k \in T}\Delta x_k < \frac{\epsilon}{4^i}$$We would get:$$\sum\limits_{k \in T}\Delta x_k < \frac{\epsilon}{2^i}$$It is clear that from the first inequality we would get:$$\sum\limits_{k \in T}\Delta x_k < \frac{1}{2^i}$$However, if we suppose his deduction to be true, then we would need:$$\forall \ \epsilon > 0 : \frac{\epsilon}{2^i} < \frac{1}{2^i} \text{ or } \frac{\epsilon}{2^i} > \frac{1}{2^i}$$Since both of them do not hold, then I would assume the author has simply made an error here. It is easily fixed once we replace:$$ \omega_f(x) \geq \frac{\epsilon}{2^i} $$with:$$ \omega_f(x) \geq \frac{1}{2^i} $$Please let me know if I am missing something.


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