Let $f$ be a function with real values, defined and continuous on $\mathbb{R}^{+}$, and admitting a finite limit at $+\infty$.
a) Show that $f$ is bounded on $\mathbb{R}^{+}$.
b) Deduce that the function $g$ defined on $\mathbb{R}^{+}$ by $\forall t \in \mathbb{R}^{+}, g(t)=t e^{\gamma t}$ where $\gamma$ is a strictly negative real, is bounded on $\mathbb{R}^{+}$.