For example we have the two sequences
$A(n) = \dfrac{3n^2+n+2}{n^2-n+1}$
and
$ B(n) = 1+(-1/2) + (-1/2)^2 + (-1/2)^3 +\cdots+ (-1/2)^{n-1}$
I can't figure out the steps to work out the upper bound of these two sequences. I've searched so much but I couldn't find any clear examples of that type of questions about bounded sequences.
For $A(n)$, you probably know (or can believe) that $A(n)\to 3$ as $n\to\infty$. Hence $A(n)\le4$ for large $n$. More precisely, write $A(n) = 3+\dfrac{4n-1}{n^2-n+1}$ and find $N$ such that $4n-1\le n^2-n+1$ for all $n\ge N$. Then $A(n)\le 4$ for all $n\ge N$.
For the first, it may help to look at $A(n+1) - A(n)$ and show that $A(n)$ is decreasing for $n \ge 2$.
For the second, do you know a formula for the sum of a geometric series?
Generally, you can make a conjecture about the upper bound of the sequence (try looking at it like a regular function and taking the limit). For example, you can guess that $ A(n) \rightarrow 3 $ as $ n \rightarrow \infty $. Then you must show, by definition of the upper bound of a sequence, that
$ \forall\epsilon>0 $, $ \exists s(\epsilon)\in A(n): inf[A(n)]-s(\epsilon)<\epsilon$
,where $ inf[A(n)] $ is the upper bound. Does that make intuitive sense? If you take $ \epsilon $ to be given, it will help you to draw a picture on the number line to see exactly what you're looking for. As a further hint, it will help you to know that there is an Archimedean property that says $ \forall\epsilon\in\mathbb{R},\exists n\in\mathbb{N}:\frac{1}{n}<\epsilon $
