Can someone provide me with a more intuitive definition of the following statement? I'm taking analysis, and the following statement pops up quite a bit: $... \lt \delta$, for some $\delta \in \Bbb R$.
Specifically, I may have $d(x_0, x_1) \lt \delta$ or $|f(z_1) - f(z_0)| \lt \delta$. 
I always understood it to mean that $x_0$ and $x_1$ were infinitesimally close to each other, as with $f(z_1)$ and $f(z_0)$.
Is my understanding correct? If yes, why do we use this particular notation? Wouldn't it be a lot easier to just say that $x_0 - x_1 \approx 0$? If not, what is the significance of this bound? 
 A: There are no infinitesimals in standard analysis. There are only numbers which can be made arbitrarily small, but usually to do so you have to make some other quantity small as well. For example, for a continuous function between two metric spaces, you can make $d(f(x),f(y))$ less than any given positive number $\varepsilon$ by taking $d(x,y)$ to be less than a number $\delta$ (depending on $\varepsilon$ and $x$). At all times these distances are either positive real numbers or zero, they are never infinitesimal.
A: The point of all this stuff is to express intuitively appealing but imprecise concepts in a form which is sufficiently precise to be used in mathematical proofs.  For example, you might think of a continuous function as one whose graph consists of an unbroken curve without gaps, jumps or missing points.  And there is nothing wrong with this concept, as long as you realise that the words "unbroken", "gaps", "jumps" and "missing points" are not really clear and need to be supplemented (not replaced) by more accurate terminology.  Thinking about examples should (with the help of your instructor if necessary) lead you to realise that these ideas are encapsulated by the definition
$$\lim_{x\to a}f(x)=f(a)\ .$$
In the case you ask about, it would indeed be much simpler to say something like "$x_1-x_0\approx 0$", and there is nothing wrong with conceptualising it in this way.  But you should also recognise that to say a quantity is "approximately equal" to $0$ is not precise (how close does it have to be before we all agree that it is approximately equal?) and that this concept needs to be formulated more rigorously.  If you try to do so, I suspect that you will end up with something very much like all that business with inequalities, epsilons and deltas.
