Muddiest Point 10/12/17

One of the things we did in this class was introduce and go through the proof for Theorem 4.2.3 (Sequential Criterion for Functional Limits). The theorem states: Given a function f:A \rightarrow \mathbb R and c \in L(A), the following are equivalent:

i.) lim_{x \to c} f(x)=L

ii.)\forall sequences (x_n) \subset A such that x_n \neq c and (x_n) \rightarrow c, it follows that (f(x_n)) \rightarrow L. For me, I think that the muddiest point lies in fully understanding the logic in the proof.

For me, I think that the muddiest point lies in fully understanding the logic in the proof, so I’m going to go through the intuition needed to understand the proof. For the \Rightarrow direction of the proof, we assume part i and prove part ii by using the topological definition of a function limit. We start with \varepsilon >0, and then say that there exists a V_\delta (c) that contains all x’s that are not equivalent to c. From the definition, it follows that f(x) \in V_\varepsilon (L). At this point, it is important to distinguish what we still need to show. We want to show that f(x_n) \in V_\varepsilon (L), so we can see that in order to bridge the gap between what we have and what we want, we need N for which the statement is true with n \geq N so that f(x_n) \in V_\varepsilon (L) is true.

The other direction of the proof is a proof by contraposition. We assume that ii is true and negate i, so it says $latex lim_{x \to c} f(x) \neq L.$ Now, we can use the topological definition of a functional limit to help us here. From the negation of ii, we know that there exists a \varepsilon_o >0 such that \forall \delta >0, \exists x \in V_\delta (c) \setminus \{c\} such that f(x) \notin V_{\varepsilon_o} (L). Since we want to contradict our negation, we proceed by picking a specific \delta and a point x_n. These are used to show that f(x_n) \notin V_{\varepsilon_o} (L). However, this means that x_n \neq c, so f(x_n) does not converge to L. This contradicts ii, which was assumed to be true, so we see that i holds to be true.

Overall, the proof of Theorem 4.2.3 is easier to complete when the topological definition of a function limit is fully understood. Also, I found that the graph on the top of page 116 in the book especially useful when trying to visualize what is happening with the proof.

Posted on by Shivani Patel
Categories: Muddiest Point

One Response

  1. Jeremy LeCrone says:
    October 20, 2017 at 9:40 am

    Shivani, thank you for your thoughts on this muddiest point, but I should point out that there are several issues in your explanation that still need some attention.

    First, the statement “there exists a V_\delta (c) that contains all x’s that are not equivalent to c” does not capture the logic of the proof. As stated, you seem to be concluding that V_\delta(c) contains ALL points in R that are not equal to c, but this is certainly not the case.

    Second, you should review the structure of a proof by contraposition. This is different from a proof by contradiction, which appears to be what you are outlining in your post. Pay careful attention to what is actually concluded in the book (and notes), and review the definition of the contrapositive of a logical implication.

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