basic sequence is equivalent to the unit vector basis of $C_0$

Question

I study from "ِ sequences and series in Banach spaces" by j.Diestel A series $\sum_n x_n$ is said to be weakly unconditionally Cauchy (wuC) if, given any permutation $\pi$ of the natural numbers, $(\sum_{k=1}^n x_{\pi(k)})$ is a weakly Cauchy sequence; alternatively, $\sum_n x_n$ is wuC if and only if for each $x^* \in X^*,$ $\sum_n |x^* x_n|<\infty.$

Theorem The following statements regarding a formal series $\sum_n x_n$ in a Banach space are equivalent:

1. $\sum_n x_n$ is wuC.
2. There is a $C > 0$ such that for any $(t_n) \in l_\infty$ $\sup_n \|\sum_{k=1}^n t_k x_k\| \leq C \sup_n|t_n|.$
3. For any $(t_n) \in c_0,$ $\sum_n t_n x_n$ converges.

Proof If we suppose 2 holds and let $(t_n) \in c_0,$ then keeping $m<n$ and letting both go off to $\infty,$ we have $\|\sum_{k=m}^n t_k x_k\| \leq C \sup_{m \leq k < n} |t_k| \rightarrow 0$

Corollary A basic sequence for which $inf_n \|x_n\| > 0$ and $\sum_n x_n$ is wuC is equivalent to the unit vector basis of $c_0.$

I have two questions Q1 I understand the steps of the proof of the corollary but i don't know how this proof proves the corollary? What i really need to prove to make sure that i prove he statement of the corollary?

Q2 in the proof of the theorem how he take cauchy series like this?