  For a state $\rho_{A_1^n B}$, we call a sequence of states $(\sigma_{A_1^k
B}^{(k)})_{k=1}^n$ an approximation chain if for every $1 \leq k \leq n$,
$\rho_{A_1^k B} \approx_\epsilon \sigma_{A_1^k B}^{(k)}$. In general, it is not
possible to lower bound the smooth min-entropy of such a $\rho_{A_1^n B}$, in
terms of the entropies of $\sigma_{A_1^k B}^{(k)}$ without incurring very large
penalty factors. In this paper, we study such approximation chains under
additional assumptions. We begin by proving a simple entropic triangle
inequality, which allows us to bound the smooth min-entropy of a state in terms
of the R\'enyi entropy of an arbitrary auxiliary state while taking into
account the smooth max-relative entropy between the two. Using this triangle
inequality, we create lower bounds for the smooth min-entropy of a state in
terms of the entropies of its approximation chain in various scenarios. In
particular, utilising this approach, we prove approximate versions of the
asymptotic equipartition property and entropy accumulation. In our companion
paper, we show that the techniques developed in this paper can be used to prove
the security of quantum key distribution in the presence of source
correlations.
