I have problems to understand the following iterative argument. I don't understand, how the denominators arrise, i.e. how the facultys build. When I plug in the inequality into itself, I get what stands in the second line, but without the denominators...
Any help is highly appreciated, thank you.
Assume you have constants $c$ and $d$. Take the following iteration. $$f^{0}(t)=cd$$ $$f^{(k+1)}(t)\leq cd+c\int_{0}^tf^{(k)}(s)\,\mathrm{d}s,$$ Assume
$$f^{(k)}(t)\leq cd\sum_{l=0}^{k}\frac{(ct)^l}{l!}$$ holds.
Then, \begin{eqnarray} f^{(k+1)}(t)&\leq& cd+c\int_0^t f^{(k)}(s)\,\mathrm{d}s\\ &\leq& cd+c\int_0^t cd \sum_{l=0}^k\frac{(cs)^l}{l!}\,\mathrm{d}s\\ &\leq& cd+cd\sum_{l=0}^kc\frac{c^lt^{l+1}}{l!(l+1)}\\ &\leq& cd+cd\sum_{l=1}^k\frac{(ct)^{l}}{(l)!}\\ &\leq&cd\left(1+\sum_{l=1}^{k+1}\frac{(ct)^{l}}{l!}\right)\\ &\leq&cd\sum_{l=0}^{k+1}\frac{(ct)^{l}}{l!} \end{eqnarray}