Also according to the definition of a positive infinitesimal:
a hyperreal number b is positive infinitesimal if b is positive but less than every positive real number.
So how are real numbers other than 0 able to have infinitesimals around them on the hyperreal line?
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I'm not sure if I understand your question.. in your title you ask what is "finite but not infinitesimal" -- in that case, for example, 1 is finite but not infinitesimal.
1 is not an infinitesimal because it is not positive and less than all positive real numbers, nor is it negative and more than all negative numbers, and nor is it 0.
Now for the question you asked in the post -- how do numbers other than 0 have infinitesimals around them? They don't, if 0 is the "0" on the number line. But if some other number became the "neutral 0" on a number line, then it could have infinitesimals around it.
EDIT: I should not write answers right before I fall asleep!
Sorry if my wording wasn't clear! What I meant was something similar to the other answer -- you can pick an arbitrary number, $x$, and you will have infinitesimals around it. You have $x \pm infinitesimal$. However in this case, $x$ itself is not an infinitesimal, but it does have infinitesimals around it.
This example was most likely referring to the ordinary number line, which has 0 in the middle. Thus, in that number line, 0 is infinitesimal, and has infinitesimals around it.
Well, I suppose that if $a\in\mathbb{R}$ and $\eta$ is a positive infinitesimal then the hyperreal number $a+\eta$ is bigger than $a$ but smaller than any real bigger than $a$.
Yet $a+\eta$ is neither real nor infinitesimal.
The situation is analogous to that of complex numbers: any complex number is a sum of a real number and an imaginary number, but (most) complex numbers are neither real, nor imaginary.