\begin{align} E \left[ \exp (tX) \right] &= \sum_{x=0}^n \exp(tx) \cdot {}_n C_x p^x (1-p)^{n-x} \\ &= \sum_{x=0}^n \ {}_n C_x \left( p e^t \right)^x (1-p)^{n-x} \\ &= \left( 1 - p + p e^t \right)^n \end{align}
\begin{align} E \left[ \exp (tY) \right] &= \sum_{y=0}^\infty \exp(ty) \cdot \exp(- \lambda) \frac{\lambda^y}{y!} \\ &= \exp(- \lambda) \sum_{y=0}^\infty \frac{\left( \lambda e^t \right)^y}{y!} \\ &= \exp(- \lambda) \cdot \exp \left( \lambda e^t \right) \\ &= \exp \left( \lambda \left( e^t - 1 \right) \right) \end{align}
(a) より、 \begin{align} E \left[ \exp \left( tZ_n \right) \right] &= \left( 1 + \frac{\lambda \left( e^t - 1 \right)}{n} \right)^n \end{align} なので、 \begin{align} \lim_{n \to \infty} E \left[ \exp \left( tZ_n \right) \right] &= \exp \left( \lambda \left( e^t - 1 \right) \right) \end{align} である。