The question is as follows:

Let $f$ be a monomorphism from a lattice $L$ to a lattice $M$.Show that $L$ is isomorphic to a sublattice of $M$.

My attempt:

Since $f$ is a monomorphism from a lattice $L$ to a lattice $M$ therefore it will be a lattice homomorphism which is injective.

Since, $f$ is injective therefore every element of $L$ will be mapped into a unique element in $f(L)$ and hence $M$,and also every element of $f(L)$ will have a unique preimage.

Thus $f:L -> f(L)$ is bijective and hence a lattice homomorphism.

Further $f(L)$ is a sublattice of $M$ and hence $L$ is isomorphic to a sublattice of $M$.

Am I correct ?