Proving f=0 on [a,b] with Analysis

[ren_101]

Homework Statement

$$f$$ is continuous on [a,b]
$$f_{1}(x)=\int^x_a f(t)dt$$
$$f_{2}(x)=\int^x_a f_{1}(t)dt$$
...
$$\forall x\in[a,b],\exists n$$ depends on x , such that $$f_{n}(x)=0.$$
prove that $$f\equiv0.$$

Homework Equations

mathematical analysis

The Attempt at a Solution

copy the taylor theorem 's proof?
but I get nothing.

 

[HallsofIvy]

If, for some n, f_n is identically 0, then it is a constant and so its derivative is identically equal to 0. But, by the fundamental theorem, the derivative of $f_n(x)= \int_a^x f_{n-1}(t)dt$ is f_n-1(x). Therefore, if f_n(x) is identically 0 on [a, b], so is f_n-1(x).