Suppose $A$ is a positive-definite matrix and $b$ is a vector
which satisfies $ b\leq \mbox{diag}(A)$ for all entries of $b$, i.e. $b_i= b^T e_i\leq e_i^T A e_i $.
The linear equations holds: $Ax=b$ where $x$ is a vector.
The question is to prove that the sum of the entries of $x$ is bounded between $0$ and $1$ $$0 \leq \sum_{i=1}^n{x_i} \leq 1.$$
Thank you very much.