The intensity function is defined as
$$lambda^*(t)=frac{f(tH_{t_n})}{1F(tH_{t_n})}$$
where $f$ is the density function and $F$ is the distribution function, and $H_{t_n}$ is the history of all the previous points of $t$ up to $t_n$.
Moreover, there is proven that $F(tH_{t_n})$ is also given as:
$$F(tH_{t_n})=1e^{int_t^{t_n}lambda^*(s)ds}$$.
An assumption is then made, saying that

$lambda^*(t)$ is nonnegative and is integrable on every interval after $t_n$.

$int_t^{t_n}lambda^*(s)ds to 1$ for $t to infty$.
It is then said that hence the three points follows:
 $0 leq F(tH_{t_n}) leq 1$
 $F(tH_{t_n})$ is a nondecreasing function of $t$.
 $F(tH_{t_n}) to 1$ for $t to infty$
Can someone explain to me how these three points follow given the two assumptions above? Thank you.