$$ A := \begin{pmatrix} 1 & \frac{1}{2} & \frac{1}{3} & \cdots & \frac{1}{n}\\ \frac{1}{2} & \frac{1}{2} & \frac{1}{3} & \cdots & \frac{1}{n}\\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & \cdots & \frac{1}{n}\\ \vdots & \vdots & \vdots & \ddots & \frac{1}{n}\\ \frac{1}{n} & \frac{1}{n} & \frac{1}{n} & \frac{1}{n} & \frac{1}{n} \end{pmatrix}$$

How to prove that all the eigenvalues of $A$ are less than $3 + 2 \sqrt{2}$?

This question is similar to this one.

I have tried the Cholesky decomposition $A = L^{T} L$, where

$$L^{T} = \left(\begin{array}{ccccc} 1 & 0 & 0 & \cdots & 0\\ \frac{1}{2} & \frac{1}{2} & 0 & \cdots & 0\\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ \frac{1}{n} & \frac{1}{n} & \frac{1}{n} & \frac{1}{n} & \frac{1}{n} \end{array}\right)$$

then

$$(L^{T})^{-1}=\left(\begin{array}{ccccc} 1 & & & \cdots\\ -1 & 2 & & \cdots\\ & -2 & 3 & \cdots\\ \vdots & \vdots & \vdots & \ddots\\ & & & -(n-1) & n \end{array}\right)$$

$$A^{-1}=L^{-1}(L^{T})^{-1}$$

How to prove the eigenvalues of $A^{-1}$

$$\lambda_{i}\geq\frac{1}{3+2\sqrt{2}}$$

Further, I find that $A$ is the covariance matrix of Brownian motion at time $1, 1/2, 1/3, \ldots, 1/n$