It's been a while, but I'll take a stab at it. these are outlines.

This statement should look familiar to you.a) For all e > 0, there exists a natural number n such that |x{n+1} − x{n}| < e

Suppose I fix n. then |x{n+1} − x{n}| is also ab) There exists a natural number n such that for all e > 0, |x{n+1} − x{n}| < efixed difference. Is it (generally) true that this difference is always less than e > 0? ((an exception is the constant sequence. x{n} = L for all n))

Suppose I fix an interval around L. Are there terms outside the interval, in general?c) There exists e > 0 such that for all natural numbers n, |x{n+1} − x{n}| < e

First consider the terms of the sequence with the highest and lowest value. you can then choose epsilon sufficiently large enough to "wrap" around all of them. If there is no upper bound and/or lower bound, what does this tell you about the sequence?d) For all natural numbers n, there exists e > 0 such that |x{n+1} − x{n}| < e