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Mathematical Analysis is a large and important branch of modern mathematics with a long history. The core notion is that of a limit, which gives the "long-term" behaviour of some process, or computes what happens as one variable gets closer and closer to a given value. The limit notion is behind many other fundamental concepts, such as continuity, derivatives, integrals, and so on. Historically, such ideas were discussed and used since at least the 17th century (for example, in connection with mathematical physics or statistics). However, they were not precisely understood; the working definitions of the day typically involved somewhat vague ideas of "infinitely small" or "infinitely large" quantities, and these can lead to confusion. We will follow in the footsteps of the pioneering mathematicians of the 19th century and formally define limits of sequences of real numbers. We will then use these definitions to rigorously deduce important properties of sequences, series and so on, which make up the bedrock of Real Analysis. Topics investigated will include: The Completeness Axiom, Sequences, Series, Absolute and Conditional Convergence of Series, Power series, Countability of sets, Continuity and properties of continuous functions, the Boundedness Theorem and the Intermediate Value Theorem.