
Monotonically Increasing
For each n $\displaystyle \in$ N let $\displaystyle f_{n}$ : [a,b] $\displaystyle \rightarrow$ R where a,b $\displaystyle \in$ R are such that a < b. Show that if F : [a,b] $\displaystyle \rightarrow$ R is such that $\displaystyle F_{n} \rightarrow $ F as n $\displaystyle \rightarrow \infty$ in a pointwise fashion for each n $\displaystyle \in$ N the function $\displaystyle F_{n}$ is monotonically increasing, then F is monotonically increasing.

Take x<y, you know that $\displaystyle F_n(x)\leq F_n(y)$. What happens when you tend to the limit as n goes to infinity?