In this subsection, $X$ is a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$, and $\left(\lambda_{i}\right)_{i \in \mathbb{N}^{*}}$ is the unique real sequence such that for all $k \in \mathbb{N}^{*}$, $k\mathbb{P}(X = k) = \sum_{j=1}^{k} j\lambda_{j} \mathbb{P}(X = k-j)$. For all $k \in \mathbb{N}^{*}$, show $$\left|\lambda_{k}\right| \mathbb{P}(X = 0) \leqslant \mathbb{P}(X = k) + \sum_{j=1}^{k-1} \left|\lambda_{j}\right| \mathbb{P}(X = k-j) \leqslant (1 - \mathbb{P}(X = 0))\left(1 + \sum_{j=1}^{k-1} \left|\lambda_{j}\right|\right)$$
In this subsection, $X$ is a random variable taking values in $\mathbb{N}$ such that $\mathbb{P}(X = 0) > 0$, and $\left(\lambda_{i}\right)_{i \in \mathbb{N}^{*}}$ is the unique real sequence such that for all $k \in \mathbb{N}^{*}$, $k\mathbb{P}(X = k) = \sum_{j=1}^{k} j\lambda_{j} \mathbb{P}(X = k-j)$.
For all $k \in \mathbb{N}^{*}$, show
$$\left|\lambda_{k}\right| \mathbb{P}(X = 0) \leqslant \mathbb{P}(X = k) + \sum_{j=1}^{k-1} \left|\lambda_{j}\right| \mathbb{P}(X = k-j) \leqslant (1 - \mathbb{P}(X = 0))\left(1 + \sum_{j=1}^{k-1} \left|\lambda_{j}\right|\right)$$