In this link, the following result is presented without proof:
Let $a, b, c$ be the lower off diagonal, diagonal, and upper off diagonal elements of a tridiagonal matrix. A pivotless LU decomposition exists if $|a_i| + |c_i| < |b_i|$.
I'm trying to convince myself that this is true (so I don't necessarily need a formal proof), but I'm not sure where to start. Or are there better existence criteria for LU decompositions on tridiagonal matrices?