  Fekete's lemma shows the existence of limits in subadditive sequences. This
lemma, and generalisations of it, also have been used to prove the existence of
thermodynamic limits in statistical mechanics. In this paper it is shown that
the two variable supermultiplicative relation $$ p_{n_1}(m_1)\,p_{n_2}(m_2)
\leq p_{n_1+n_2}(m_1 + m_2) $$ together with mild assumptions, imply the
existence of the limit $$ \log \mathcal{P}_\#(\epsilon) = \lim_{n\to\infty}
\frac{1}{n} \log p_n(\lfloor \epsilon n \rfloor).$$ This is a generalisation of
Fekete's lemma. The existence of $\mathcal{P}_\#(\epsilon)$ are proven for
models of adsorbing walks and polygons, and for pulled polygons. In addition,
numerical data are presented estimating the general shape of $\log
\mathcal{P}_\# (\epsilon)$ of models of square lattice self-avoiding walks and
polygons.
