Master of Science / Department of Mathematics / Natalia Rojkovskaia / The topic of limits and sequences run through the math syllabi from high school to graduate school. However rigorous proofs of this concept is not seen up until a students second year in college or even later. This text is aimed at presenting proofs of limits of sequences at a level accessible to high school and undergraduate students who are interested in learning such proofs. The ''Epsilon-N" definition of the limit with proofs using this definition is presented in the text. We also look at properties of limits of sequences and their proofs as well as sequences without limits (that is sequences that diverge). We include some graphical representations of some sequences which can help one to determine whether a sequence will converge or diverge. Finally, the text contains a good number of exercises for the reader, some solved and others with hints to direct the reader in solving them.