  The long standing Lech's conjecture in commutative algebra states that for a
flat local extension $(R,\mathfrak{m})\to (S,\mathfrak{n})$ of Noetherian local
rings, we have an inequality on the Hilbert--Samuel multiplicities: $e(R)\leq
e(S)$. In general the conjecture is wide open when $\dim R>3$, even in equal
characteristic. In this paper, we prove Lech's conjecture in all dimensions,
provided $(R,\mathfrak{m})$ is a standard graded ring over a perfect field
localized at the homogeneous maximal ideal. We introduce the notions of lim
Ulrich and weakly lim Ulrich sequences. Roughly speaking these are sequences of
finitely generated modules that are not necessarily Cohen--Macaulay, but
asymptotically behave like Ulrich modules. We prove that the existence of these
sequences imply Lech's conjecture. Though the existence of Ulrich modules is
known in very limited cases, we construct weakly lim Ulrich sequences for all
standard graded domains over perfect fields of positive characteristic.
