This is an interactive website to help you prepare for the cumulative portion of the final exam. You should be able to **flawlessly write the kinds of proofs presented below**. Pay careful attention to the templates. You must **also study the homework for the last third of the course**. The final exam will be longer than previous exams. *Before you click on the dotted boxes, try to write the proofs youself.*

We prove the theorem with \(E := \{a, b\}\), as the general case can be obtained using this result and the Additivity Theorem.

Let \(\varepsilon > 0\) be given. Because *any* tagged partition with then . If the subintervals in \(\dot{P}\) are \([x_{i-1}, x_i]\), then applied to \(F\) on \([x_{i-1}, x_i]\) implies that there exist such that , \(1 \leq i \leq n\). Now let's use these \(u_i\) as tags and the endpoints we've already examined to obtain the tagged partition , which satisfies \(\left|\left|\dot{P}\right|\right| < \delta\) by assumption, and we compute \(S(f;\dot{P_u})\). If we add the terms and evaluate the telescoping sum, we find . Finally, we substitute into to obtain , and because \(\varepsilon\) was arbitrary, the proof is complete. \(\Box\)

Given \(\varepsilon > 0\), let

. Then if \(n \geq K\), we have . Thus, we find that \((\tfrac{1}{n}) \rightarrow 0 \) by Definition 2.1.5. \(\Box\)Given \(\varepsilon > 0\), let

. We may assume that . Then if \(n>K\), we have . Thus, we learn that \((\tfrac{n^2}{3n^2+2}) \rightarrow \tfrac{1}{3} \) by Definition 2.1.5. \(\Box\)Given \(\varepsilon > 0 \), let

. Then if \(m > n \geq H\), we have . Thus, \(x_n\) is Cauchy by Definition 2.6.4. \(\Box\)Given \(\varepsilon > 0 \), let

. Then if \(m > n \geq H\), we have . Thus, \(y_n\) is Cauchy by Definition 2.6.4. \(\Box\)Given \(\varepsilon > 0 \), let

, assuming \(\delta < 1\). Then if , we compute that by the lemma. Thus, we find . Therefore, we obtain \(\lim\limits_{x \to 1} (x^2 + 3x) = 4\) by Definition 3.1.6. \(\Box\)Given \(\varepsilon > 0 \), let

. Then if , we have . Thus, \(f(x)\) is continuous at \(x=3\) by Definition 4.1.2. \(\Box\)Given \(\varepsilon > 0 \), let

. Then if , we have . Thus, \(f\) is uniformly continuous by Definition 4.3.3. \(\Box\)We compute that

. Thus, we learn that by Definition 5.1.1. \(\Box\)Given \((x_n) \rightarrow x \) and \((y_n) \rightarrow y \), we will show that \((x_n + y_n) \rightarrow x + y \). Since

, given \(\varepsilon > 0 \) there exists a such that if \(n \geq K_1 \) then . Similarly, because , there exists a such that if \(n \geq K_2 \) then . Let . Then if , we have . Thus, we learn that \((x_n + y_n) \rightarrow x + y \) by definition. \(\Box\)Given \(\varepsilon > 0\), there exists a

such that if then . Thus, if , we have . Therefore, we know that a convergent sequence is Cauchy. \(\Box\)