Proving sequences are Cauchy in a metric space

In this question I have concluded that the sequences in b) and c) are both Cauchy with respect to $d$. This is because I deduced that $f(a_1 + \ldots + _n) \geq min\{f(a_1), \ldots, f(a_n)\}$ and used telescoping in both questions. In part c), I deduced that $f(b_{n+1} - b_n) \geq 2n + 1$ for $n \geq 2$, but I'm not sure this is correct. I was just wondering if I have mis-understood this question and my approach is wrong? What about d), what should I be doing? Thank you.