Word Count : 481

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To verify the hypothesis and conclusions of Fatou's Lemma for the sequence of functions \( \{f_n\} \), we need to examine the behavior of the functions and their pointwise limit. Let's first clarify the functions given: The sequence \( f_n(x) \) is defined as: \[ f_n(x) = \begin{cases} 2n & \text{if } \frac{1}{2n} < x < \frac{1}{n}, \\ 0 & \text{if } x \in \left(0, \frac{1}{2n}\right) \cup \left(\frac{1}{n}, 1\right). \end{cases} \] ### Step 1: Pointwise Limit of \( f_n(x) \) To analyze the pointwise limit of \( f_n(x) \), we need to observe what happens to ________ __________ _________ _____ ______ _________ _____ ___ _________ __________ ___.
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