I am taking my first analysis course and the notion of limits implying convergence or not is some what blurry.
Can you think of a sequence $s_k$ for which $s_{k+1} - s_k$ goes to 0 but for which $s_k$ doesn't converge? and if so why is the existence of that limit not sufficient for convergence?
Convergence depends crucially on the space the sequence takes values:
Pick $s_n = 1/n$ in $(0,1)$. Notice that as $n \rightarrow \infty$, then the distance between two terms is getting smaller and smaller, but $s_n$ is not converging to a point in $(0,1)$. This is an example of what is called a Cauchy sequence. Note that in particular spaces it may be the case that cauchy sequences do not converge. In the real line this is impossible since $\mathbb{R}$ is complete.
Another example: Take the space of rational numbers $\mathbb{Q}$. Try to see what I mean by taking a sequence with rational numbers $q_n$ that converge to an irrational number.