Convergence, supremum and limit Hey guys I am kind of confused about these three terms.
How to prove if a sequence is convergent? 
How to prove the sup S or inf S exists?
How to prove the limit of a sequence exists?
How to show that a sequence is bounded?
To show that the limit exist, we can show that the sequence is decreasing/increasing and bounded. What about the rest? Thanks.

A sequence $x_n$ converges to some limit $L$ iff for every $\epsilon>0$, there is a $N\in\mathbb{N}$ such that if $n>N$, then $|x_n-L|<\epsilon$; that is, for any arbitrarily small positive $\epsilon$, you can find some natural number such that past that number, all the terms of the sequence are less than $\epsilon$ from the limit.
Alternatively, you can prove $x_n$ converges if $\forall\epsilon>0$, there is some natural number $N$ such that $m,n>N$ implies that $|x_n-x_m|<\epsilon$. This is called a Cauchy sequence. For real numbers, if a sequence is Cauchy, that is equivalent to saying it converges, so you can prove that a series has a limit even if you don't know what it is.
If you want to show that a sequence $x_n$ is bounded, just show that there is some $N$ such that $|x_n|\leq N$, for all $n$.
The supremum is just the smallest upper bound of $x_n$; likewise, the infimum is the largest lower bound.
