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\newcommand{\myCourseNumber}{Math 142A}
\newcommand{\myAssignmentDueDate}{Tuesday, February 24, 2025}
\newcommand{\myAssignmentTitle}{Homework Assignment 4}
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\begin{document}

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\begin{questions}

\question Show that if \ $\sum a_n$ \ and \ $\sum b_n$ \ are convergent series of nonnegative numbers, then \ $\sum \sqrt{a_n \, b_n}$ \ converges.

\bigskip

\question Find a series \ $\sum a_n$ \ which diverges by the Root Test but for which the Ratio Test gives no information.

\bigskip

\question Let $(a_n)$ be a sequence of nonzero real numbers such that the sequence \ $\left(\frac{a_{n+1}}{a_n}\right)$ \ is a constant sequence.  Show that \ $\sum a_n$ \ is a geometric series.

\bigskip

\question Let \ $(a_n)_{n \in \mathbb{N}}$ \ be a sequence such that \ $\lim\inf |a_n| = 0$.  Prove there is a subsequence \ $\left( a_{n_k} \right)_{k \in \mathbb{N}}$ \ such that \ $\displaystyle \sum_{k=1}^{\infty} a_{n_k}$ \ converges.

\bigskip

\question \begin{parts}
\part Exhibit an example of a divergent series \ $\sum a_n$ \ for which \ $\sum a_n^2$ \ converges.
\part Show that if \ $\sum a_n$ \ is a convergent series of nonnegative terms, then \ $\sum a_n^2$ \ also converges.
\part Exhibit an example of a convergent series \ $\sum a_n$ \ for which \ $\sum a_n^2$ \ diverges.
\end{parts}

\bigskip

\question Prove that if $(a_n)$ is a decreasing sequence of positive real numbers and if \ $\sum a_n$ \ converges, then \ $\displaystyle \lim_{n \to \infty} n a_n = 0$

\bigskip

%%Fitzpatrick 3.1.9
\question Suppose that the function \ $f : \mathbb{R} \to \mathbb{R}$ \ is continuous at the point \ $x_0$ \ and that \ $f(x_0) > 0$. \ Show that there is an interval \ $I_n = \left( x_0 - \frac{1}{n}, \ x_0 + \frac{1}{n} \right)$ \ for some \ $n \in \mathbb{N}$ \ for which \ $f(x) > 0$ \ for every \ $x \in I_n$.

\bigskip

%%Fitzpatrick 3.1.13
\question A function \ $f : D \to \mathbb{R}$ \ is said to be \emph{Lipschitz} provided that there is a number \ $C \geq 0$ \ with
\begin{displaymath} |f(u) - f(v)| \leq C \, |u - v| \quad \text{for all $u$ and $v$ in $D$}. \end{displaymath}
Show that a Lipschitz function is continuous.

\bigskip

%%Fitzpatrick 3.1.14
\question Suppose the function \ $\lambda : \mathbb{R} \to \mathbb{R}$ \ has the the property that $\lambda(u + v) = \lambda(u) + \lambda(v)$ \ for all $u, v$.
\begin{parts}
\part Define the number $m$ by $m := \lambda(1)$.  Show that \ $\lambda(x) = m \, x$ \ for all rational numbers $x$.
\part Show that if $\lambda$ is continuous, then \ $\lambda(x) = m \, x$ for all $x \in \mathbb{R}$.
\end{parts}

\bigskip

\question Let $f$ be a real-valued function whose domain is a subset of $\mathbb{R}$.

Show that $f$ is continuous at \ $x_0 \in \text{dom}(f)$ \ if and only if for every sequence $(x_n)$ in \ $\text{dom}(f) \setminus \{x_0\}$ converging to $x_0$, we have \ $\displaystyle \lim_{n \to \infty} f(x_n) = f(x_0)$.

\end{questions}

\end{document}