If a matrix is symmetric, tridiagonal, and diagonally dominant, is it positive definite? If a matrix $A \in \mathbb{R}^{N\times N}$ is symmetric, tridiagonal, diagonally dominant, and all the diagonal elements of $A$ are positive, then is $A$ also positive-definite? If it is, how to prove it? The proposition looks to be true according to a statement in this question, Is a symmetric positive definite matrix always diagonally dominant? and a comment to it. However, I can't come up to a proof by myself. I would appreciate your help.

You don't need tridiagonal. Gerschgorin's theorem (plus the fact the eigenvalues of a symmetric matrix are real) implies that all eigenvalues of a strictly diagonally dominant symmetric matrix with positive diagonal elements are positive, and all eigenvalues of a diagonally dominant symmetric matrix with positive diagonal elements are nonnegative.

You do need that "strictly", e.g. $\pmatrix{1 & 1\cr 1 & 1\cr}$ is diagonally dominant but not strictly diagonally dominant, and has an eigenvalue $0$.