Let $A,D \in \mathbb{R}^{n\times n}$ be two positive definite matrices given by
$$ D = \begin{bmatrix} 2 & -1 & 0 & 0 & \dots & 0\\ -1 & 2 & -1 & 0 & \dots & 0\\ 0 & -1 & 2 & -1 & \dots & 0\\ \vdots & \ddots & \ddots & \ddots & \ddots & 0\\ 0 & \dots & 0 & -1 & 2 & -1\\ 0 & 0 & \dots & 0 & -1 & 2 \end{bmatrix}, \quad A = \begin{bmatrix} 2 c_{1,1} & -c_{1,2} & 0 & 0 & \dots & 0\\ -c_{2,1} & 2 c_{2,2} & -c_{2,3} & 0 & \dots & 0\\ 0 & -c_{3,2} & 2 c_{3,3} & -c_{3,4} & \dots & 0\\ \vdots & \ddots & \ddots & \ddots & \ddots & 0\\ 0 & \dots & 0 & -c_{n-1,n-2} & 2 c_{n-1,n-1} & -c_{n-1,n}\\ 0 & 0 & \dots & 0 & -c_{n,n-1} & 2 c_{n,n} \end{bmatrix} $$$$ D = \begin{bmatrix} 1 & -1 & 0 & 0 & \dots & 0\\ -1 & 2 & -1 & 0 & \dots & 0\\ 0 & -1 & 2 & -1 & \dots & 0\\ \vdots & \ddots & \ddots & \ddots & \ddots & 0\\ 0 & \dots & 0 & -1 & 2 & -1\\ 0 & 0 & \dots & 0 & -1 & 1 \end{bmatrix}, \quad A = \begin{bmatrix} c_{1,2} & -c_{1,2} & 0 & 0 & \dots & 0\\ -c_{2,1} & c_{2,1} + c_{2,3} & -c_{2,3} & 0 & \dots & 0\\ 0 & -c_{3,2} & c_{3,2} + c_{3,4} & -c_{3,4} & \dots & 0\\ \vdots & \ddots & \ddots & \ddots & \ddots & 0\\ 0 & \dots & 0 & -c_{n-1,n-2} & c_{n-1,n-2} + c_{n-1,n} & -c_{n-1,n}\\ 0 & 0 & \dots & 0 & -c_{n,n-1} & c_{n,n-1} \end{bmatrix} $$ with $c_{i,j} \in (0,c_+]$$c_{i,j} = c_{j,i} \in (0,c_+]$ for all $i,j=1,\dots,n$ for a $c_+ \in (0,\infty)$.
I would like to prove that independent of the dimension of $n$ $$x^\top A x \le c_+ x^\top D x$$ holds for all $x\in \mathbb{R}^n$. If this is not the case does there exist a counter example?
This is somehow related to that the norms of the induced scalar products of the matrices $A$ and $D$ are equivalent with factor $c_+$.