MIT's 18.100B Real Analysis, taught by Tobias Holck Colding, continues with the formal treatment of sequences. Colding defines what it means for a sequence of real numbers to converge, working through the epsilon-based definition and proving the basic properties that follow from it, such as uniqueness of limits and boundedness of convergent sequences. He introduces the notion of a subsequence and shows how subsequences relate to the convergence of the original sequence. The lecture runs 79 minutes on the blackboard, building proofs step by step rather than skipping to results, consistent with the rest of the course's rigorous approach to real analysis. This is lecture four in the series, assuming familiarity with the basic definitions and notation for sequences and limits introduced earlier in the course.