Suppose $c \geqslant 1$ and $X_{c}$ is a positive real-valued random variable with probability density

$$f_{c}(t)=A_{c} t^{c-1} e^{-t^{c}},$$

for $t>0$, where $A_{c}$ is a constant.

Find the constant $A_{c}$ and show that, if $c>1$ and $s, t>0$,

$$\mathbb{P}\left[X_{c} \geqslant s+t \mid X_{c} \geqslant t\right]<\mathbb{P}\left[X_{c} \geqslant s\right]$$

[You may assume the inequality $(1+x)^{c}>1+x^{c}$ for all $x>0, c>1$.]