قالب LaTeX: ورقة مسائل رياضيات

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{\color{white}
  \noindent{\large\bfseries MATH 301 --- Real Analysis}\hfill
  {\normalsize Problem Set 4}\\[3pt]
  \noindent{\small Due: Friday, March 7\quad
  \textcolor{rulecol}{|}\quad
  Show all work for full credit}
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\vspace{1.6cm}

% ── Problems ──────────────────────────────────────────────────────────────────
\begin{tcolorbox}[probstyle,title={Problem 1}]
\textbf{Cauchy sequences.}\quad
Let $(x_n)_{n\ge 1}$ be a Cauchy sequence in $\R$.
Prove that $(x_n)$ converges, carefully establishing
both the existence of the limit and that the sequence attains it.

\medskip
\textit{Hint: First show that every Cauchy sequence is bounded, then
extract a convergent subsequence via the Bolzano--Weierstrass theorem.}
\end{tcolorbox}

\vspace{8pt}

\begin{tcolorbox}[probstyle,title={Problem 2}]
\textbf{Integrability and absolute value.}\quad
Let $f:[a,b]\to\R$ be Riemann integrable.
\begin{enumerate}[label=(\alph*),leftmargin=*,nosep,itemsep=4pt]
  \item Prove that $|f|$ is Riemann integrable on $[a,b]$.
  \item Establish the \emph{triangle inequality for integrals}:
    \[
      \left|\int_a^b f(x)\,dx\right| \;\le\; \int_a^b |f(x)|\,dx.
    \]
\end{enumerate}
\end{tcolorbox}

\vspace{8pt}

\begin{tcolorbox}[probstyle,title={Problem 3}]
\textbf{Pathological example.}\quad
Construct an explicit function $f:\R\to\R$ such that:
\begin{itemize}[leftmargin=*,nosep,itemsep=3pt]
  \item $f$ is continuous \emph{nowhere}, yet
  \item $|f|$ is continuous \emph{everywhere}.
\end{itemize}
Prove that your construction has the claimed properties.
\end{tcolorbox}

\vspace{8pt}

\begin{tcolorbox}[probstyle,title={Problem 4}]
\textbf{Absolute convergence.}\quad
Let $(a_n)_{n\ge 1}$ be a sequence of real numbers.
\begin{enumerate}[label=(\alph*),leftmargin=*,nosep,itemsep=4pt]
  \item Prove: if $\displaystyle\sum_{n=1}^\infty|a_n|<\infty$,
    then $\displaystyle\sum_{n=1}^\infty a_n$ converges.
  \item Determine whether the converse holds. If false, provide a counterexample.
\end{enumerate}
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