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\begin{document}

\section{Questions}

\textbf{Q1:} Let $\{f_n\}_{n=2}^\infty \subset C[0,1]$ be a sequence of piecewise linear functions, where $n\in\mathbb{N}$ and $n\geq 2$, and each function $f_n$ is defined by:
\[
f_n(x) = \begin{cases}
    1, & \text{for } x \in \left[0, \frac{1}{2} - \frac{1}{n} \right], \\
    \text{linear}, & \text{for } x \in \left[ \frac{1}{2} - \frac{1}{n}, \frac{1}{2} + \frac{1}{n} \right], \\
    0, & \text{for } x \in \left[ \frac{1}{2} + \frac{1}{n}, 1 \right]
\end{cases}
\]

\begin{itemize}
    \item Show that the sequence $\{f_n\}_{n=2}^\infty$ is Cauchy in the $d_2$ metric, where
    \[
    d_2(f,g) = \left( \int_0^1 |f(x) - g(x)|^2 \, \mathrm{d}x \right)^{1/2}, \quad f, g \in C[0,1].
    \]
    \item Show that the sequence $\{f_n\}_{n=2}^\infty$ is not Cauchy in the $d_\infty$ metric, where
    \[
    d_\infty(f,g) = \sup_{x \in [0,1]} |f(x) - g(x)|, \quad f, g \in C[0,1].
    \]
\end{itemize}

\textbf{Q2:} Show that $\ell^2$ is a vector space; that is, if $x, y \in \ell^2$, then $x + y \in \ell^2$ and $\lambda x \in \ell^2$ for any $\lambda \in \mathbb{R}$. You may assume, without proof, the triangle inequality for the norm $\| \cdot \|_2$ on $\R^n$ for any $n\in \N$.

\textbf{Q3:} Show that the subset $c_{00}$ is dense in the metric space $\ell^2$.

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