Question: Mark each statement True or False.

1. Every interior point of \(S \subset \mathbf{R}\) is an accumulation point of \(S\).
2. Every bounded sequence is convergent.
3. Every Cauchy sequence is convergent.
4. Let \(f: D \rightarrow \mathbf{R}\). If \(f\) does not have a limit at \(c\), then there exists a sequence \(\left\{s_{n}\right\}\) in \(D\) with each \(s_{n} \neq c\) such that \(\left\{s_{n}\right\}\) convergences to \(c\), but \(f\left(s_{n}\right)\) is divergent.

(c) If \(f\) is continuous on \(D\) and \(D\) is closed and bounded, then \(f(D)\) is closed.

(f) If \(f\) is continuous on \((a, b)\), then \(f\) is uniformly continuous.

(g) If \(f\) and its first \(n\) derivatives are continuous on [a, b] and differentiable on \((a, b)\). Let \(x_{0} \in(a, b)\), then for all \(x \in[a, b]\), we have

\(f(x)=f\left(x_{0}\right)+f^{\prime}\left(x_{0}\right)\left(x-x_{0}\right)+\frac{f^{\prime \prime}\left(x_{0}\right)}{2 !}\left(x-x_{0}\right)^{2}+\cdots+\frac{f^{(n)}\left(x_{0}\right)}{n !}\left(x-x_{0}\right)^{n}\)

(h) Let \(f\) be integrable on $[a, b]$ and let \(g\) be the function obtained from \(f\) by changing only a finite values. Then \(g\) is also integrable and \(\int_{a}^{b} f(x) d x=\int_{a}^{b} g(x) d x\)

1. If \(f\) is bounded on $[a, b]$, then \(f\) is integrable on $[a, b] \(.

(j) Let\) f$ be defined on \([a, \infty) .\) Assume that \(f\) is integrable on $[a, c]$ for every \(c>a\). Then \(\int_{a}^{\infty} f(x) d x\) exists (i.c. the improper integral is convergent).

Price: $2.99

Solution: The downloadable solution consists of 2 pages
Deliverable: Word Document