Suppose $X$ is a Banach space, $\left \{ e_n \right \}$ a sequence in $X,\ \overline {\text{span}\left \{ e_n \right \}}=X$ such that $e_n\neq 0$ and such that there is a constant $K$ with the property that $\left \| \sum_{n=1}^{N}x_ne_n \right \|\le K\left \| \sum_{n=1}^{M}x_ne_n \right \|$ whenever $N<M$ and $\left \{ x_n \right \}$ is any sequence of scalars. The clam is then that $\left \{ e_n \right \}$ is a basis for $X$. Carothers has a proof (on p. 31), which I found non-intuitive. Below is my attempt. Is it correct? Can someone explain Carother's proof? In particular, the statement "We now only need to show that.." on the same page as that cited above.
Let $Y=\text{span}\left \{ e_n \right \}$ and define the projections $P_n:Y\to Y$ in the usual way by $\sum_{n=1}^{N}x_ne_n\mapsto \sum_{n=1}^{n}x_ne_n;\ N\ge n.$ Then, $\left \| P_n \right \|\le K$ so $P_n$ extends by continuity to the whole of $X$.
Now define for $Y\ni x=\sum_{n=1}^{N}x_ne_n$ the functional $f_n:Y\to \mathbb F$ by $f_n(x)=x_n,\ $ which is well-defined since $e_n\neq 0$, and, again by continuity, extends to the whole of $X$. Thus, we can write, for any $x\in X,\ P_n(x)=\sum_{i=1}^{n}f_i(x)e_i.$
Now, let $x\in X$ and $Y\ni x_n\to x$. Let $\epsilon >0$ and choose $J$ such that $\left \| x-x_J \right \|<\epsilon .$ Also, choose $N$ so large that $P_N (x_J) = x_J.$ Then, $\left \| \sum_{n=1}^{N} f_n(x)e_n-x\right \|=\left \| P_N(x)-x\right \|\le \left \| P_N(x)-P_N(x_J) \right \|+\left \| P_N(x_J)-x_J \right \|+\left \| x_J-x \right \|\le K\epsilon+0+\epsilon=(K+1)\epsilon.$
Uniqueness follows easily because if $\sum x_ne_n=\sum f_n(x)e_n,\ $ then $f_n(\sum x_ne_n)=f_n(\sum f_n(x)e_n) \Rightarrow x_n=f_n(x).$