Prove or disprove $(1-y)^x(1+xy)<1$ where $x>1$, $0<y<1$.

Question

Let$$x>1$$ $$0<y<1$$ Is it possible to prove or disprove this following: $$(1-y)^x(1+xy)<1$$ I tested many sample results and could not find a case that make it false yet. I think Bernoulli's inequality might help.

Answer 1

Using Bernoulli's Inequality, $$(1-y)(1+xy)^{1/x}\leqslant (1-y)(1+y)=1-y^2<1$$

Raise both sides to the power $x$ to get your desired inequality.