Let $F_n$ denote the $n$th polynomial obtained through Gram-Schmidt orthogonalization applied to the sequence $$\{ 1, x, x^2, x^3 , \cdots , x^n \}$$ with the inner product $$ \int_0^1 f(x) \cdot g(x) dx.$$ I am trying to show that $x \cdot F_n$ is a linear combination of $\{ F_{n−1}, F_n, F_{n+1} \}$ for every positive integer $n$.
I tried using mathematical induction by assuming that this statement holds true for all $n = k$ and wanted to show this holds true for $n = k +1$. But I am having a hard time using the inductive hypothesis.
The polynomial $F_n$ satisfies $\deg F_n=n$ and is orthogonal to all polynomials of degree strictly less than $n.$ Hence for $k\le n-2$ we get $$\langle xF_n,F_k\rangle =\int\limits_0^1xF_n(x)F_k(x)\,dx=\int\limits_0^1F_n(x)[xF_k(x)]\,dx=\langle F_n,xF_k\rangle =0$$ i.e. $xF_n$ is orthogonal to every polynomial $F_k$ for $k\le n-2.$ Thus $xF_n$ is a linear combination of $F_{n-1},F_n$ and $F_{n+1}.$