I am puzzled how did Bartle get $$|g_k|\leq\sum_{j=k}^\infty |g_{j+1}-g_j|$$ (second last line)?
I tried using Triangle Inequality and ended up with one extra term:
$$\begin{align*} |g_k|&=|g_k-g_{k+1}+g_{k+1}-g_{k+2}\dots+g_{k+N-1}-g_{k+N}+g_{k+N}|\\ &\leq\sum_{j=k}^{N-1}|g_{j+1}-g_j|+|g_{k+N}| \end{align*}$$
Even after taking limits $N\to\infty$, the extra term $|g_{k+N}|$ does not necessarily go to zero.
Thanks for any help!
I am not sure about the inequality you mention, but since $|g_k| \le g$ seems to be sufficient for your purposes, you can get this by \begin{align*} |g_k| &= |g_1+(g_2-g_1)+\dots+(g_k-g_{k-1})| \\ &\le |g_1|+|g_2-g_1|+\dots|+|g_k-g_{k-1}|\\ &=|g_1| + \sum_{j=1}^{k-1} |g_{j+1}-g_j| \\ &\le |g_1| + \sum_{j=1}^\infty |g_{j+1}-g_j|=g \end{align*} I.e., you can use the same sum as in your book, but the range is not $j\ge k$, but $j<k$.