Meaning of the numbers in a sequence definition 
The sequence $(a_n)$ tends to $+ \infty \iff$ given any number $C$, there's a number $N$ such that $n > N \implies a_n \ge C.$

Given a certain $N$ it's not difficult to prove the implication, but what's the meaning of $n, N, C$? Do they stand for something concrete other than being numbers(all real?)?
 A: $n$ and $N$ are natural numbers and are the index of the terms $a_n$ of the sequence.
The definition says that for any $C$ you can find a "suitable" $N$ such that from the "term" $a_N$ of the sequence on (this means: $n > N$), all the terms $a_n$  are greater-or-equal to $C$.
A: The sequence $(a_n)_{n\geq1}$ is given term for term, and there is nothing arbitrary about that. The numbers $C$ and $N$ however appearing in the definition of the statement $\lim_{n\to\infty} a_n=\infty$ are "arbitrary", in other words: $C$ and $N$  serve only as variable names for a certain logical implication. 
We want to express the idea that "for large $n$ all $a_n$ are really large" in a logically stringent way, and saying that for any $C$ however large it is guaranteed that $a_n>C$ as soon as $n$ is large enough, i.e., larger than some $N=N(C)$, is the definitive way of formulating our intended meaning.
A: One can think of a series that "tends to infinity" in the following way:
for all $x \in \mathbb{R}$, there is some $N \in \mathbb{N}$ so that $a_n> x$. This just means the sequence gets as big as we want it to eventually. But sequences in some sense don't have to diverge in one particular way. We often conceive of them as monotone/ always increasing, but this is just some bias we have. For example, if you take the sequence defined by 
$ a_n = \begin{cases} 
      \ n  \textrm{ if $n$ is odd} \\
       \ 0  \textrm{ otherwise} \\
   \end{cases}$
This of course doesn't "tend to infinity" in the usual sense of the word, but it is true that for each $x \in \mathbb{R}$ we can find some $N$ so that $a_n$ is bigger than it. Here, the sequence indeed diverges to infinity.
Logically: $\{a_n\}$ diverges to $\infty_+$  $\iff$ $(\forall x \in \mathbb{R})(\exists N \in \mathbb{N})(a_N>x)$
More technically, we can think of a sequence in the reals as follows:
$f: \mathbb{N} \to \mathbb{R}$, where $\{a_n\}$ denotes the set $\{f(1), f(2),...\}$ in the usual ordering. 
Then, on a cartesian plane, take any function $f: \mathbb{R}_+ \to \mathbb{R}$ and just look at the x-values for $x \in \mathbb{N}$, and consider the restriction of $f$ to these values.
Notice that the domain here matters.
So, I guess the take away is that diverging to infinity is certainly something specific, and there are some different ways of thinking about it. The important part to me, is that it only talks about what happens eventually. A sequence can be $a_i=0$ for all $i \leq 10^{10}$, and then start increasing in increments of $10^{-10}$-- but it will still "tend to infinity" (it will also be monotone! even this is well-behaved.)
