Given a sequence formula_15, another sequence formula_25 is a subsequence of formula_15 if formula_27 for all positive integers formula_28 and formula_29 is a strictly increasing sequence of natural numbers. Roughly speaking, a limit is the value that a function or a sequence "approaches" as the input or index approaches some value. (This value can include the symbols formula_30 when addressing the behavior of a function or sequence as the variable increases or decreases without bound.) The idea of a limit is fundamental to calculus (and mathematical analysis in general) and its formal definition is used in turn to define notions like continuity, derivatives, and integrals. (In fact, the study of limiting behavior has been used as a characteristic that distinguishes calculus and mathematical analysis from other branches of mathematics.) The concept of limit was informally introduced for functions by Newton and Leibniz, at the end of the 17th century, for building infinitesimal calculus. For sequences, the concept was introduced by Cauchy, and made rigorous, at the end of the 19th century by Bolzano and Weierstrass, who gave the modern ε-δ definition, which follows. Definition. Let formula_31 be a real-valued function defined on formula_32. We say that formula_33 tends to formula_34 as formula_35 approaches formula_36, or that the limit of formula_33 as formula_35 approaches formula_36 is formula_34 if, for any formula_41, there exists formula_42 such that for all formula_43, formula_44 implies that formula_45. We write this symbolically as formula_46, or formula_47.Intuitively, this definition can be thought of in the following way: We say that formula_48 as formula_49, when, given any positive number formula_50, no matter how small, we can always find a formula_51, such that we can guarantee that formula_33 and formula_34 are less than formula_50 apart, as long as formula_35 (in the domain of formula_31) is a real number that is less than formula_51 away from formula_36 but distinct from formula_36. The purpose of the last stipulation, which corresponds to the condition formula_60 in the definition, is to ensure that formula_47 does not imply anything about the value of formula_62 itself. Actually, formula_36 does not even need to be in the domain of formula_31 in order for formula_65 to exist.