I think the $\epsilon$-$\delta$ definition is difficult to the students who don't know where the definition comes from. I have once tried to teach a student by asking what it means for a sequence to converge and let him make up his own definition. I did give a lot of hints here and there, but I made him think through it. It was a lengthy discussion but fun. Later, the $\epsilon$-$\delta$ definition was built informally by converting "all sequences" into an interval of length $\delta$ (or $2\delta$ if you prefer). This conversion was understood pretty quickly, probably because the concept is the same.
I: How do you define convergence of a series?
Student: It's when the number tends toward something.
I: How do you define "tends toward something"?
Student: ... It gets closer and closer to some number.
I: Ok, so if I have this sequence: $1, 0, \frac 12, 0, \frac 13, 0, \frac 14, 0, \ldots$, does it converge?
Student: It does because it eventually goes to zero.
I: But $\frac 14$ is farther away from zero than the term before it.
Student: ...But afterwards, everything is smaller than $\frac 14$.
And the discussion goes on. These are not exact words, but the content is close to what I actually discussed.