The proof I followed is under the following link
The problem is described as follow:
- By steps listed in the proof the first lemma illustrated: If $g$ is a continuous function on $[0,1]$ and $\int_0^1 g(x)x^ndx=0$ for $n=0,1,2,…$ then $g(x)=0$ for $x\in[0,1]$.
- The lemma 2 is to prove that if $\mathcal{L}\{f(x)\}=0$ than $f(x)=0,x\in[0,\infty)$. By definition of laplace transform: \begin{aligned} \mathcal{L}\{f(x)\}&=\int_0^\infty e^{-st}f(t)dt\\ &=\int_1^0 u^{s}f(-\ln(u))\left(-\frac{1}{u}\right)du\\ &=\int_0^1 u^n u^{s-1-n}f(-\ln(u))du\\ &=0\\ \end{aligned}
- However the domain of $u$ is $(0,1]$ which means that it is an improper integral instead of a proper integral with the domain of $[0,1]$. Which mean that I cannot apply the result from lemma 1 to prove it.
Hence, if anyone can help me on proving the improper integral version of lemma 1 or, provide another prove. Thanks so much!!