Word Count : 481

---

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 __________ ________ ____ ______ ________ ______ __________ ___ ___.
__________ ______ ___ ____ _______ ______ _________ ________ _____.
_____ _______ ___ ___ __________ ________ ________ ________ ________ _________ _________ _______.
_____ __________ ____ _____ ______ ___ ________ ___.
__________ ____ ________ ______ _____ ______ _____.
_____ ____ ______ ________ _____ ________.
____ _________ __________ __________ _________ __________ __________ ______ _________ ___ ______.
_______ ____ _____ _____ _________ ________ ___ _______ ____ __________ _________.
____ _________ ______ ____ ________ ________ _________ _________ __________ ____ _________ ___.
___ __________ ____ __________ _______ ______ ____ _________ __________ _______ ______ _________.
____ ________ _____ __________ ________ ____.
_______ _____ _______ _______ _____ ________ ________.
____ _________ ___ _________ _______.
_________ _________ _____ _____ ____ ___ ________.
__________ _____ ____ __________ ____ __________.
__________ ________ _____ ________ _______ ______.
______ _________ _____ ____ _______.
________ _________ __________ ___ _______ _______ __________ ________ _____ __________ _________.
______ _________ ___ ___ _______.
_____ ___ ___ _____ _______ ________ _______ ________.
__________ ________ __________ ________ ________ ______ _________ ___.
_____ __________ __________ __________ _________ ________ __________ _________ ______.
_____ ______ _______ _______ _____ _______ ____ _________.
______ ________ _____ _________ _________ ________ ________ _____.
____ ______ _______ ___ _____ ____ ________ ____ ______ ______.
______ ____ _________ _______ _____ _______.
__________ ____ _____ _____ ______ _____ ______.
________ __________ _________ _______ ____ ___ ______ _________ ______ ______.
________ _______ _________ ______ ____ ____ __________ __________ ______ ______ ______.
___ ______ ________ _____ ______ ________ ___ _______ ____.
_______ ___ ________ ________ ________ ______ ______.
_________ ___ __________ __________ ______ ____ ___ _____ _____ _______.
___ _______ __________ ___ _____ _________ __________ _________ ____ _______ __________ ________.
__________ __________ _______ ________ _______ _________.
_____ ____ ___ ______ _________ ________ ______ ___ ___.
_____ __________ _____ _________ ____ _____ ______ ___.
_____ _____ ___ ____ ______.
____ ______ _______ __________ ____ ___ ____ __________ __________.
_______ _________ __________ ______ ______ ____ _____ ____ _____ _________.
___ ____ _______ ________ __________ ____ _____ ___.
_________ ______ ___ _______ _________ ___.
____ ________ ________ __________ ________ __________ _______.
___ ______ _____ ____ _________.
_____ ___ _______ __________ ____ ______ ____ _______ ______.
______ _______ _______ _______ _______.
__________ _________ _______ _________ ________ __________ ______ ________ _________.
___ _________ ___ _______ ___ __________ __________ ________ ______ ________ ____.
Get Full Answer on WhatsApp