real analysis – Prove that $int_0^1g=sum_{j=1}^Nlambda_jg(x_j)$,$forall gin V$.

Suppose $Vsubset C(0,1)$is a linear space and $dim V=N$,and there exists an element $uin V$,such that $int_0^1uneq 0$.Suppose $Omega subset (0,1)$is a closed set ,and if $fin V$,$sup_{Omega}fleq 0$,then $int_0^1fleq 0$.Prove that there exists $x_1,cdots,x_Nin Omega$,and $lambda_1,cdots,lambda_Ngeq 0$,so that
begin{align*}
int_0^1g=sum_{j=1}^Nlambda_jg(x_j ),forall gin V
end{align*}

Below is what I can thought.Try to solve this problem by induction on $dim V$.

When $N=1$,$V=L(f_1)$.then $int_0^1f_1neq 0$, Without loss of generality´╝îwe can suppose $int_0^1f_1>0$.Then by the property of $Omega$ we have
$$sup_{Omega}f_1>0.$$
So there exists $x_1in Omega$,such that $f_1(x_1)>0$.We take$lambda_1=frac{int_0^1f_1}{f_1(x_1)}$,then $lambda_1>0$,and
$$int_0^1f_1=lambda_1f_1(x_1).$$
So the conclusion holds for $N=1$.