If f and g are two increasing functions such that gof is defined, then

Let $x_1, x_2 \in R$ such that $x_1<x_2$

$\Rightarrow f\left(x_1\right)<f\left(x_2\right)$         (∵  f is increasing)

$\Rightarrow g\left(f\left(x_1\right)\right)<g\left(f\left(x_2\right)\right)$          (∵  g is increasing)

$\Rightarrow(gof)\left(x_1\right)<(gof)\left(x_2\right)$

⇒ gof is increasing.