We denote by $(f_{n})_{n \geqslant 1}$ the sequence of functions defined on $]0, +\infty[$ by:
$$f_{n}(t) = \begin{cases} \left(1 - \frac{t}{n}\right)^{n} t^{x-1} & \text{if } t \in ]0, n[ \\ 0 & \text{if } t \geqslant n \end{cases}$$
Show that for all integers $n$, $n \geqslant 1$, the function $f_{n}$ is continuous and integrable on $]0, +\infty[$.