Exercises 5.8 Week 6 Exercises, due 2 November 2021
Let \((a_n)_{n=0}^{\infty}\) be a bounded sequence of real numbers. For every \(k\in\NN_0\text{,}\) we define
\begin{equation*}
b_k = \inf \{a_n : n \geq k\}
\end{equation*}
and
\begin{equation*}
c_k = \sup \{a_n : n \geq k\}\text{.}
\end{equation*}
1.
Prove that the sequence \((b_k)_{k=0}^{\infty}\) converges. An analogous argument will show that \((c_k)_{k=0}^{\infty}\) converges. We write
\begin{equation*}
\liminf_{n\to\infty} a_n = \lim_{k \to \infty} b_k
\end{equation*}
and
\begin{equation*}
\limsup_{n\to\infty} a_n = \lim_{k \to \infty} c_k\text{.}
\end{equation*}
2.
For every \(n\in\NN\text{,}\) let \(x_n = (-1)^n(1-1/n)\text{.}\) Compute \(\liminf_{n\to\infty} x_n\) and \(\limsup_{n\to\infty} x_n\text{.}\)
3.
Show that
\begin{equation*}
\liminf_{n\to\infty} a_n = \limsup_{n \to \infty} a_n
\end{equation*}
if and only if \((a_n)_{n=0}^{\infty}\) converges.