prove this series is convergent prove that the series $(a_1-a_2)+(a_2-a_3)+(a_3-a_4)+...$ converges if and only if the sequence $\{a_n\}$ converges.
I know a series is convergent if the sequence of its partial sums is bounded. I do not know how to apply that to this question.

Hints:
Note that the series you have is
$$\sum_{k=1}^\infty (a_k - a_{k+1})$$
It's partial sum, in turn, is
$$\sum_{k=1}^n (a_k - a_{k+1})$$
Consider trying a few small values for $n$, and consider the pattern that emerges. Use this to make a formula that is equivalent to the partial sum. In the context of that formula, what can you say happens to it as $n \to \infty$, both when $(a_n)$ is a divergent sequence, and when it is a convergent sequence?
(We approach this issue this way because we say $\sum_{k=1}^\infty b_k$ converges only if the sequence of partial sums - $\sum_{k=1}^n b_k$, for $n=1,2,3,\cdots$ - converges. Your definition of convergence for infinite sums isn't quite correct.)
