Answer
$\{a_{n}\}$ is bounded and monotonic
Work Step by Step
Bounded Sequences:
1. A sequence $\{a_{n}\}$ is bounded above when there is a real number $M$ such that $a_{n}\leq M$ for all $n$.
The number $M$ is called an upper bound of the sequence.
2. A sequence $\{a_{n}\}$ is bounded below when there is a real number $N$ such that $N\leq a_{n}$ for all $n$.
The number $N$ is called a lower bound of the sequence.
3. A sequence $\{a_{n}\}$ is bounded when it is bounded above and bounded below.
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The $\displaystyle \frac{1}{n}$ term is at most 1, never zero so the sequence is bounded with
upper bound M=4
lower bound N=3
$\{a_{n}\}$ is bounded.
Checking for monotony:
$\displaystyle \frac{1}{n} > \frac{1}{n+1}\qquad \times(-1)$
$-\displaystyle \frac{1}{n} < -\frac{1}{n+1}\qquad +4$
$4-\displaystyle \frac{1}{n} < 4-\frac{1}{n+1}$
$a_{n} < a_{n+1}$
$\{a_{n}\}$ is monotonic
(increasing, therefore nondecreasing)
Graph (see below) confirms our conclusions.
