Problem: in each of the following cases , give an example of a real valued sequence $\{x^k\}$ meeting the stated properties.
a) $\limsup_{k\to\infty}$$\{x^k\}$$=$$+\infty$=$\liminf_{k\to\infty}\{x^k\}=+\infty$
b) $\limsup_{k\to\infty}$$\{x^k\}$=+$\infty$ $\liminf_{k\to\infty}\{x^k\}=-\infty$
c) $\limsup_{k\to\infty}$$\{x^k\}$=+$\infty$ $\liminf_{k\to\infty}\{x^k\}$=0
d) $\limsup_{k\to\infty}$$\{x^k\}$= 0 $\liminf_{k\to\infty}\{x^k\}=-\infty$
I am unsure as to how to solve these problems. They were left as an exercise for the reader in my lecture. Can anyone breakdown (or solve) these specific problems? They have left me really stumped.
a) $x^{k}=k$ for all $k$
b) $x^{k}=k$ for all even $k$ and $x^{k}=-k$ for all odd $k$
c) $x^{k}=k$ for all even $k$ and $x^{k}=\frac 1 k$ for all odd $k$
d) $x^{k}=-\frac 1 k$ for all even $k$ and $x^{k}=- k$ for all odd $k$