Real Analysis 12 | Examples for Limit Superior and Limit Inferior [dark version]

Limit superior (lim sup) and limit inferior (lim inf) always exist for any real sequence, taking values in the extended real line (real numbers or ±∞). A key takeaway from the opening example is that these two quantities can land anywhere in that extended range: for the sequence a_n = (-1)^n n (−1, 2, −3, 4, −5, …), the even-index subsequence diverges to +∞ while the odd-index subsequence diverges to −∞. That forces lim sup a_n = +∞ and lim inf a_n = −∞, showing that the set of possible values for lim sup and lim inf includes both infinities.

For convergent sequences, lim sup and lim inf collapse to the same real number. If a sequence converges to L, then both lim sup and lim inf equal L because there is only one accumulation value. Conversely, if lim sup and lim inf are equal and finite, there is only one accumulation value, so the sequence must converge to that common value. The same “equality criterion” extends to divergence toward infinity: if the sequence diverges to +∞, then lim sup and lim inf both equal +∞ (and they match as extended-real values, not as finite numbers). Similarly, divergence toward −∞ corresponds to lim sup = lim inf = −∞. Another structural fact used repeatedly in calculations is monotonic ordering: lim sup is always the largest accumulation value, so lim sup ≥ lim inf. In particular, if lim inf = +∞ then lim sup must also be +∞; and if lim sup = −∞, then lim inf must be −∞.

The most practical part comes next: inequalities that relate lim sup/lim inf of sums and products to those of the individual sequences. For two sequences a_n and b_n, one generally has lim sup (a_n + b_n) ≤ lim sup a_n + lim sup b_n, with a crucial caveat about undefined expressions like +∞ + (−∞). The inequality can fail to be meaningful when the right-hand side involves such indeterminate forms, but when the extended-real arithmetic is well-defined, the inequality holds. For products, a similar inequality holds under non-negativity assumptions: lim sup (a_n b_n) ≤ (lim sup a_n)(lim sup b_n), because sign changes can reverse inequalities; again, expressions like 0·(+∞) are treated as undefined and must be handled carefully.

The same style of inequalities applies to lim inf, but with reversed inequality directions. The transcript emphasizes that these are not just abstract rules: concrete examples demonstrate both the direction and when inequalities become strict. For instance, with a_n = 1, −1, 1, −1, … and b_n = 0, 2, 0, 2, …, the sum is constantly 1, so lim sup and lim inf of the sum are 1, while lim sup a_n + lim sup b_n = (1) + (2) = 3 and lim inf a_n + lim inf b_n = (−1) + (0) = −1, giving strict inequalities. A second example modifies a_n to keep it non-negative for the product case, producing a constant product sequence and again showing the inequality can be strict.

Overall, the session’s goal is to equip learners with a small set of reliable extended-real inequalities for lim sup and lim inf—plus the exceptions needed when ±∞ arithmetic becomes indeterminate—so these bounds can be used confidently in later limit computations.