In the book R. M. Young, An introduction to non-harmonic Fourier series, I came across the following problem (page 18):


**Problem.** Show that the sequence $\left \{ \frac{1}{x+1},\frac{1}{x+2},\frac{1}{x+3}, \dots \right \}$ is complete in $L^2[0,1]$.


I tried to apply the Müntz-Szasz theorem but it didn't work out. Any ideas or approaches how to show completeness here? Thanks!



Completeness of the sequence $\left \{ \frac{1}{x+1},\frac{1}{x+2},\frac{1}{x+3}, \dots \right \}$ in $L^2[0,1]$
Let $S$ be the Hilbert subspace of $L^2[0,1]$ spanned by the sequence. For every positive integer $n$, the function $\frac{n}{x+n}=\frac{1}{1+x/n}$ lies in $S$. For $n$ large, this function is very close to the constant $1$ function in sup-norm, hence $1\in S$. Now we prove by induction that, for every nonnegative integer $m$, the functions $x^m$ and $\frac{x^m}{x+n}$ ($n=1,2,\dotsc$) lie in $S$. We have already established the base case $m=0$. So let us assume that $m>0$ and the functions $x^{m-1}$ and $\frac{x^{m-1}}{x+n}$ ($n=1,2,\dotsc$) lie in $S$. Then $x^{m-1}-\frac{nx^{m-1}}{x+n}=\frac{x^m}{x+n}$ lies in $S$. Moreover, $\frac{nx^m}{x+n}=\frac{x^m}{1+x/n}$ lies in $S$ as well, so by the same approximation argument as before, $x^m\in S$.


It follows that every polynomial lies in $S$, therefore $S=L^2[0,1]$ by the Weierstrass approximation theorem.


