Is "$\prec$" a formal math symbol?

Question

Is "$\prec$" a formal symbol?

I've never seen it before.

Answer 1

If you are asking about what the symbol is typically used for, and what sort of precedence there is for the use of it, then the following would be my answer:

Generally when $\prec$ is used, it denotes some type of abstract order relation, see https://en.wikipedia.org/wiki/Order_theory.

Like in your above example, we give some order to the function by defining an order relation given by the convergence criteria.

Answer 2

In mathematics, you are allowed to introduce any symbol you like, and to define that symbol, and to apply that definition. So, for example, perhaps I wish to introduce the notation $@\!!\!\!\!<$ for a binary relation on the natural numbers. Then I have to tell you the definition of that relation:

$m \, @\!!\!\!\!< \, n$ means that $m^2 + 3n$ is a prime number.

Great!

And silly, yes... I can't dream of what it could be applied to.

But my point is, that's how notations and their definitions work throughout mathematics.

If you look at the screen shot in your own post, you'll see that's exactly what's going on: that passage is introducing the notation $\prec$ for a binary relation on the set of functions of $n$, and it is telling you the definition of that relation:

$f(n) \prec g(n)$ means that $\lim_{n \to \infty} \frac{f(n)}{g(n)} = 0$.

And there are many important applications of this relation.