Laplace Final value theorem proof

Question

I was studing FVT and the proof I used was this : $$\int_0^\infty \hat f(t)e^{-st}\,dt=sF(s)-f(0)$$ $$\lim_{s\to0}\int_0^\infty \hat f(t)e^{-st}\,dt=\lim_{s\to0}[sF(s)]-f(0)$$ $$\int_0^\infty \hat f(t)\,dt=\lim_{s\to0}[sF(s)]-f(0)$$ $$f(\infty)-f(0)=\lim_{s\to0}[sF(s)]-f(0)$$ $$f(\infty)=\lim_{s\to0}[sF(s)]$$ it looks straightforward, but i've heard about a condition : " The standard assumptions for the final value theorem require that the Laplace transform have all of its poles either in the open-left-half plane (OLHP) or at the origin, with at most a single pole at the origin. In this case, the time function has a finite limit." now i'm just wondering regarding the proof where does this come from ? why is this condition needed? What is the reason behind it ?