Let $L$ be a locally finite (every interval in $L$ is finite) and graded (there exists a rank function $\rho:L\rightarrow \mathbb{Z}$ such that $\rho(x)=\rho(y)+1$ whenever $x$ covers $y$) lattice.
Is it true that for every finite (induced) sublattice $S$ of $L$ ($S$ preserves the meet and join of $L$) the distance between vertices in the (underlying undirected graph of the) Hasse diagram of $S$ is equal to the distance between them in the Hasse diagram of $L$?
A simple proof/counter example would be appreciated.
Edit: As @William Elliot answered below, a sublattice does not preserve distances, I intended $S$ to be convex... Let $S$ be some interval $[u,v]$ in $L$ (it can be shown that it is a sublattice as needed). Is the distance preserved in this case?
Consider the lattice 0 < a,b < 1 and the sublattice {0,1}.