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Despite tremendous amount of research on the design of low-density parity-check (LDPC) codes with belief propagation decoding over different types of binary-input output-symmetric memoryless (BIOSM) channels, most results on this topic are based on numerical methods and optimization which do not provide much insight into the design process. In particular, systematic design of provably capacity achieving sequences of LDPC code ensembles over the general class of BIOSM channels, has remained a fundamental open problem. For the case of the binary erasure channel, explicit construction of capacity achieving sequences have been proposed based on a property called the flatness condition. In this paper, we propose a systematic method to design universally capacity approaching rate-compatible LDPC code ensemble sequences over BIOSM channels. This is achieved by interpreting the flatness condition over the BEC, as a successive maximization (SM) principle that is generalized to other BIOSM channels to design a sequence of capacity approaching ensembles called the parent sequence. The SM principle is then applied to each ensemble within the parent sequence, this time to design rate-compatible puncturing schemes. As part of our results, we extend the stability condition which was previously derived for degree-2 variable nodes to other variable node degrees as well as to the case of rate-compatible codes. Consequently, we rigorously prove that using the SM principle, one is able to design universally capacity achieving rate-compatible LDPC code ensemble sequences over the BEC. Unlike the previous results on such schemes over the BEC in the literature, the proposed SM approach is naturally extendable to other BIOSM channels. The performance of the rate-compatible schemes designed based on our systematic method is comparable to those designed by optimization.