Error analysis of a numerical time-averaging via Poisson equation

Question

I'm following the proof of the paper, Mattingly et al., Convergence of numerical time-averaging and stationary measures via Poisson equations (https://arxiv.org/pdf/0908.4450.pdf). Specifically, I'm looking into the proof of Theorem 5.6 in page 16. The part I got stuck on is the following set of inequalities; $$ |Q_k|\leq \sum_{i=k+1}^p \frac{\Delta^{i-k}}{(i+1-k)!}|Q_i| + K\Delta^{p+1-k} + \frac{K}{T}, \quad k=2,\dots, p. $$ Ignoring the context, you just have to know that $\Delta \in (0,1)$, $K$ is an arbitrary constant for upper bounding, and $T = N\Delta$ for some $N > 0$ ($N$ is the total number of steps and $\Delta$ is a step size), and $p$ is a constant integer. As far as I can see, it is trivial that $|Q_p| \leq K\Delta + K/T$, and from this, \begin{aligned} |Q_{p-1}| &\leq \frac{\Delta}{2}|Q_p| + K \Delta^2 + \frac{K}{T} := K\Delta^2 + \frac{K\Delta}{T} + \frac{K}{T},\\ |Q_{p-2}| &\leq \frac{\Delta}{2}|Q_{p-1}| + \frac{\Delta^2}{6}|Q_p| + K\Delta^3 + \frac{K}{T} := K\Delta^3 + \frac{K\Delta^2}{T} + \frac{K\Delta}{T} + \frac{K}{T}, \end{aligned} and so we have for $k=2,\dots, p$, $$ Q_k \leq K\Delta^{p+1-k} + \frac{K}{T}(\Delta^{p-k} + \dots +\Delta + 1). $$ But the paper states that "in particular, $|Q_p| \leq K\Delta + K/T$. Hence $|Q_k| \leq K\Delta^{p+1-k} + K/T$ which together with (5.27) implies (5.22)". I don't understand how we could drop the intermediate terms $\Delta^{p-k} + \dots + \Delta + 1$ when $\Delta \in (0,1)$. Even if I admit that $|Q_p| \leq K\Delta + K/T$, putting this into (5.27) in the paper gives \begin{aligned} \left|\frac{1}{N}\sum_{n=0}^{N-1}\mathbf{E}\phi(X_n)-\bar\phi \right| &\leq \sum_{k=2}^p\frac{\Delta^{k-1}}{k!}|Q_k| + K\Delta^p + \frac{K}{T} \\ &\leq \sum_{k=2}^p \frac{1}{k!}\left(K\Delta^p + \frac{K\Delta^{k-1}}{T}\right) + K\Delta^p + \frac{K}{T} \\ & := K\Delta^p + \frac{K}{T}(1 + \Delta + \dots + \Delta^{p-1}), \end{aligned} so we still have the terms $\Delta + \dots + \Delta^{p-1}$, but the final big-$O$ notation (5.22) does not include them. I'm pretty sure I'm missing something obvious here; Can anyone point out what am I missing? Thanks in advance!