can formalize this property like this: there exist subsequences and of (where and are increasing) for which we have On the other hand, there exists a so that for all To recapitulate: If is greater than the limit superior, there are at most finitely many greater than if it is less, there are infinitely many. If is less than the limit inferior, there are at most finitely many less than if it is greater, there are infinitely many. In general, The liminf and limsup of a sequence are respectively the smallest and greatest cluster points. For any two sequences of real numbers the limit superior satisfies subadditivity whenever the right side of the inequality is defined (that is, not or ): Analogously, the limit inferior satisfies superadditivity: In the particular case that one of the sequences actually converges, say then the inequalities above become equalities (with or being replaced by ). For any two sequences of non-negative real numbers the inequalities and hold whenever the right-hand side is not of the form If exists (including the case ) and then provided that is not of the form Examples As an example, consider the sequence given by the sine function: Using the fact that π is irrational, it follows that and (This is because the sequence is equidistributed mod 2π, a consequence of the equidistribution theorem.) An example from number theory is where is the -th prime number. The value of this limit inferior is conjectured to be 2 – this is the twin prime conjecture – but has only been proven to be less than or equal to 246. The corresponding limit superior is , because there are arbitrarily large gaps between consecutive primes. Real-valued functions Assume that a function is defined from a subset of the real numbers to the real numbers. As in the case for sequences, the limit inferior and limit superior are always well-defined if we allow the values +∞ and −∞; in fact, if both agree then the limit exists and is equal to their common value (again possibly including the infinities). For example, given , we have and . The difference between the two is a rough measure of how "wildly" the function oscillates, and in observation of this fact, it is called the oscillation of f at 0. This idea of oscillation is sufficient to, for example, characterize Riemann-integrable functions as continuous except