I am not getting the concepts of bounded when it comes to sequences. Can someone tell me these answers and explain why they are the answer?
Choose the best description for each sequence: Bounded from Above, Bounded from Below, Both Bounded from Above and Below, or None of These:
$a_n = \frac{(-1)^n}{n}$
$b_n = -4-3n$
$c_n = 5 + \frac{n}{2}$
$d_n = (-2)^n$
(original at https://i.stack.imgur.com/GZmt0.png)
Assuming that the $n$ is counting from 1 to $\infty$ for all four, it is clear that: ${a_n}$ is bounded both above and below, since for $n=1$, $a_1 = -1$, and that for $n=\infty$, $a_\infty \rightarrow 0$. For ${b_n}$, the sequence if bounded above by any value greater than -7, since $b_1 =-7$ and the sequence ${b_n}$ is monotonically decreasing. ${c_n}$ is bounded below, since its lowest value is $5+\frac{1}{2}$, and increases infinitely from there on. Lastly, ${d_n}$ is unbounded. We see this because $\vert {b_k} \vert > \vert {b_{k-1}} \vert$, but ${b_k} > {b_{k-1}}$ is false half of the time. In other words, its oscillates and the oscillations get larger and larger each time.