Discrete and continuous variables

Let $X$ be a random variable with distribution function $F$, and let $a = \left(a_{m}:-\infty < m < \infty \right)$ be a strictly increasing sequence of real numbers satisfying $a_{-m} \rightarrow -\infty$ and $a_{m} \rightarrow \infty$ as $m \rightarrow \infty$. Define $G(x) = \mathbb {P}\left(X \leq a_{m}\right)$ when $a_{m-1} \leq x < a_{m}$, so that $G$ is the distribution function of a discrete random variable. How does the function $G$ behave as the sequence $a$ is chosen in such a way that $\sup_{m}\left|a_{m}-a_{m-1}\right|$ becomes smaller and smaller?

Let $X$ be a random variable and let $g: \mathbb {R} \rightarrow \mathbb {R}$ be continuous and strictly increasing. Show that $Y = g(X)$ is a random variable.

Let $X$ be a random variable with distribution function

Let $F$ be a distribution function which is continuous and strictly increasing. Show that $Y = F^{-1}(X)$ is a random variable having distribution function $F$. Is it necessary that $F$ be continuous and/or strictly increasing?

Show that, if $f$ and $g$ are density functions, and $0 \leq \lambda \leq 1$, then $\lambda f+(1-\lambda ) g$ is a density. Is the product $f g$ a density function?

Which of the following are density functions? Find $c$ and the corresponding distribution function $F$ for those that are.