The question is to prove or disprove (via counterexample) the following three statements:
a) A sequence $x_n$ is Cauchy iff $$\sum_{n=1}^{\infty} x_{n+1} - x_n$$ is convergent.
I think this one is true. By the Cauchy criterion for convergence of series, $\sum_{j=n}^{m} x_{j+1} - x_j = x_m - x_{n+1}$ converges iff $x_m - x_{n+1} \leq \epsilon$ where $\epsilon > 0$ is given. This would suggest $x_n$ is a Cauchy sequence.
b) If $x_n$ is Cauchy in a metric space $(M, d)$, then $$\sum_{n=1}^{\infty} d(x_{n+1}, x_n)$$ is convergent.
This one also seems to be true. If $x_n$ is Cauchy, then for $m, n \geq N$, $d(x_n, x_{n+1}) \leq \epsilon / (m - n)$ where $\epsilon > 0$ is given. Then $\sum_{j=n}^{m} d(x_{j+1}, x_j) \leq \sum_{j=n}^{m} \epsilon / (m - n) \leq \epsilon$, which would suggest convergence of the series as well. I'm not 100% sure about this proof, though.
c) If $x_n$ are points in a metric space $(M, d)$ such that $$\sum_{n=1}^{\infty} d(x_{n+1}, x_n)$$ is convergent, then $x_n$ is a Cauchy sequence in $(M, d)$.
Not sure about this one.