Real Analysis 11 | Limit Superior and Limit Inferior [dark version]

TL;DR

Divergence to +∞ means the sequence eventually exceeds every real bound: for each M, some N satisfies a_n > M for all n ≥ N.

Briefing Cornell Notes

Briefing

Limit superior and limit inferior turn messy, non-convergent behavior of real sequences into two precise “best possible” accumulation values—largest and smallest—so analysts can still talk about long-run extremes. For a sequence (a_n) of real numbers, the key idea is that it may have many accumulation (cluster) values, possibly infinitely many, and possibly none among the real numbers. In that case, the framework also allows improper accumulation values at +∞ or −∞ (used as symbols, not as actual real numbers). This matters because it guarantees a clean way to identify the “top” and “bottom” long-term behavior even when ordinary convergence fails.

The transcript first grounds the improper cases with the standard notion of divergence to +∞: for every constant M, the sequence eventually exceeds M. Similarly, divergence to −∞ corresponds to eventually going below any bound from below. It also notes that a sequence cannot simultaneously diverge to both +∞ and −∞; only one direction can occur. When a sequence has no real accumulation points, it can still be said to “cluster at infinity” in the improper sense. With this setup, every sequence has at least one improper accumulation value when it has no real accumulation values, which then enables defining extremes among accumulation values.

From there, the definitions become crisp. The limit superior, written lim sup a_n (also denoted limsup_{n→∞} a_n), is defined as the largest accumulation value of the sequence—whether that largest value is a real number or +∞. The limit inferior, lim inf a_n, is the smallest accumulation value—whether that smallest value is a real number or −∞. These two quantities are special because they summarize the sequence’s long-run behavior using only one number (or one improper symbol) for each side.

A geometric argument explains why these definitions match a “cutoff” procedure. Consider plotting points (n, a_n) on a number line. The supremum of the tail of the sequence—meaning the largest value among terms with indices ≥ N—can only stay the same or decrease as N grows, because each new cutoff discards more early terms. That produces a monotone decreasing sequence of suprema. Its ordinary limit (as N→∞) equals the largest accumulation value, hence equals lim sup a_n. Dually, for lim inf, one looks at the infimum of the tails: the smallest value among terms with indices ≥ N. As N increases, the infima can only stay the same or increase, yielding a monotone increasing sequence whose limit equals the smallest accumulation value, hence lim inf a_n. The transcript also emphasizes that +∞ and −∞ can appear in these tail sup/inf sequences as well.

Finally, it flags that further examples and additional properties are left for the next installment, but the core takeaway is already established: lim sup and lim inf are the systematic way to extract the maximum and minimum accumulation behavior from any real sequence, convergent or not.

Cornell Notes

Limit superior (lim sup) and limit inferior (lim inf) capture the extreme long-run behavior of a real sequence using accumulation (cluster) values. Accumulation values may be real numbers, and if the sequence has no real cluster points, improper accumulation values at +∞ or −∞ are allowed as symbols. By definition, lim sup a_n is the largest accumulation value, while lim inf a_n is the smallest. A practical way to compute them uses tails: take the supremum of {a_k : k ≥ n} and let n→∞; its limit equals lim sup a_n. Similarly, take the infimum of {a_k : k ≥ n} and let n→∞; its limit equals lim inf a_n. This works even when the tail suprema/infima become +∞ or −∞.

What does it mean for a sequence to “diverge to +∞,” and why is +∞ treated as a symbol rather than a number?

Diverging to +∞ means: for every constant M, there exists an index N such that for all n ≥ N, a_n > M. The transcript stresses that +∞ is not a real number; it’s a symbolic way to express that the sequence eventually exceeds every real bound. The same idea applies in reverse for divergence to −∞, where eventually a_n falls below any chosen lower bound.

How do accumulation (cluster) values relate to convergence, and what happens when a sequence has no real accumulation points?

Accumulation values are the real numbers the sequence gets arbitrarily close to along subsequences. If a sequence has no real accumulation points (no cluster points on the real line), the framework still assigns improper accumulation values: +∞ if the sequence eventually grows beyond every bound, or −∞ if it eventually drops below every bound. This lets the theory still identify “extreme” long-run behavior.

What are the formal definitions of lim sup a_n and lim inf a_n?

lim sup a_n is defined as the largest accumulation value of the sequence, allowing that largest value to be +∞. lim inf a_n is the smallest accumulation value, allowing it to be −∞. The transcript uses these definitions to ensure every sequence has well-defined “top” and “bottom” accumulation behavior.

Why does lim sup a_n equal the limit of tail suprema sup{a_k : k ≥ n}?

Let s_n = sup{a_k : k ≥ n}. As n increases, the set {k ≥ n} shrinks, so the supremum cannot increase; s_n is monotonically decreasing (or can become +∞). The transcript’s key point is that the limit of this monotone sequence of suprema as n→∞ equals the largest accumulation value of the original sequence, which is lim sup a_n.

What is the dual relationship for lim inf a_n using tail infima inf{a_k : k ≥ n}?

Let t_n = inf{a_k : k ≥ n}. As n increases, the tail set shrinks, so the infimum cannot decrease; t_n is monotonically increasing (or can become −∞). The limit of t_n as n→∞ equals the smallest accumulation value of the original sequence, which is lim inf a_n.

Review Questions

Given a sequence with no real cluster points, how would you decide whether its improper accumulation value is +∞ or −∞?
Explain why the sequence of tail suprema is monotone (and in which direction).
Describe the tail-supremum/tail-infimum procedure that produces lim sup and lim inf. What role do +∞ and −∞ play?

Key Points

1
Divergence to +∞ means the sequence eventually exceeds every real bound: for each M, some N satisfies a_n > M for all n ≥ N.
2
Improper accumulation values at +∞ and −∞ are symbols used when a sequence has no real accumulation points.
3
The limit superior lim sup a_n is the largest accumulation value of (a_n), possibly +∞.
4
The limit inferior lim inf a_n is the smallest accumulation value of (a_n), possibly −∞.
5
lim sup a_n can be computed as the limit of the tail suprema sup{a_k : k ≥ n} as n→∞.
6
lim inf a_n can be computed as the limit of the tail infima inf{a_k : k ≥ n} as n→∞.
7
Tail suprema form a monotonically decreasing sequence, while tail infima form a monotonically increasing sequence (with +∞/−∞ allowed).

Highlights

lim sup a_n is the “largest cluster value” the sequence keeps returning to, even if that extreme is +∞.

lim inf a_n is the “smallest cluster value” the sequence keeps returning to, even if that extreme is −∞.

Cutting off more and more initial terms turns the problem into limits of monotone sequences of tail suprema/infima.

+∞ and −∞ are not numbers but symbolic endpoints that make the definitions work for non-bounded or non-clustering sequences.

Topics

Limit Superior
Limit Inferior
Accumulation Values
Improper Limits
Tail Suprema/Infima