A sequence is bounded below if all of its terms stay above some real number. In Calculus II, you use this idea when studying sequences, convergence, and whether a sequence can keep dropping forever.
In Calculus II, bounded below means a sequence has a floor: there is some real number L so that every term satisfies a_n \ge L. You do not need the sequence to touch that floor, only to stay above it. Any number smaller than or equal to all the terms counts as a lower bound.
A sequence can have more than one lower bound. For example, if all terms are at least 2, then 2 is a lower bound, but so are 1, 0, and -100. The point is not finding the only bound, because there usually is not one. The point is checking whether at least one such bound exists.
This matters in sequence problems because bounded below is a property of the whole list, not just a few early terms. If a sequence starts large and then levels off above a line, it is bounded below. If it eventually keeps decreasing without ever settling above any floor, then it is not bounded below.
A quick example is a_n = 1/n for n = 1, 2, 3, ... . The terms are 1, 1/2, 1/3, 1/4, and so on. Every term is at least 0, so the sequence is bounded below by 0. It is also bounded below by -1, -10, or any smaller number. On the other hand, a_n = -n is not bounded below because the terms keep dropping without limit, so no real floor works.
Do not mix up bounded below with convergence. A sequence can be bounded below and still fail to converge, like sin(n) shifted upward by 2 if you ever see sequences built from trig values. Being bounded below only says the terms never fall past a floor. Convergence says the terms get closer to a single limit. In Calc II, bounded below often shows up as one part of a bigger argument about monotonic sequences and convergence.
Bounded below shows up all over the sequences unit because it gives you a first check on how a sequence behaves as n gets large. If a sequence is decreasing and bounded below, Calculus II often uses that combination to conclude it converges. That is a classic theorem you will see in sequence and series work: monotonic plus bounded means the terms settle down instead of running off forever.
It also helps you diagnose whether a formula can possibly have a limit. If the terms keep falling with no floor, then the sequence is unbounded below and cannot converge to a real number. That matters when you are analyzing recursive sequences, rational expressions, exponential patterns, or any list of terms written with an index variable n.
This term also connects to how you reason from graphs and tables. If a homework problem gives you the first several terms of a sequence, you look for a consistent lower bound, not just a pattern in the size of the numbers. A sequence that seems to shrink might still stay above 0, above -3, or above some other floor depending on the rule.
In short, bounded below is one of the first structural questions you ask about a sequence. It helps separate sequences that can settle into a limit from ones that can keep falling away forever.
Keep studying Calculus II Unit 5
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view galleryConvergence
A bounded below sequence can still diverge, so bounded below is not the same thing as having a limit. In Calc II, though, bounded below often appears alongside monotonic behavior as part of a convergence proof. If the terms are decreasing and never drop past a floor, the sequence is much easier to classify.
Monotonic Sequence
Monotonic sequences move in one direction, either increasing or decreasing. Bounded below matters most for decreasing sequences, because a decreasing sequence with a floor is a strong candidate for convergence. When you see both ideas together, you are usually checking whether the terms are trapped between moving one way and staying above a lower bound.
Bounded Above
Bounded above is the mirror idea: the sequence stays below some ceiling instead of above some floor. Calc II problems often ask you to check both bounds, especially when a sequence is trapped between two values. Knowing which side matters helps you avoid mixing up a lower bound with an upper bound.
unbounded sequence
An unbounded sequence has no finite ceiling or floor in at least one direction. If a sequence is not bounded below, it may keep decreasing without limit, like -n. Spotting unbounded behavior tells you quickly that no real lower bound exists and that a real-valued limit is not possible.
A quiz or homework problem will usually ask you to decide whether a sequence has a lower bound, often from an explicit formula or a list of terms. You might test a sequence like a_n = 1/n, a_n = -n, or a recursive rule and explain what real number works as a floor. If the sequence is decreasing, bounded below is often the exact property you need before applying a convergence theorem. On written work, the safest move is to name a specific lower bound and then show that every term stays above it.
Bounded below means the sequence stays above a floor, while bounded above means it stays below a ceiling. It is easy to flip them if you are reading inequalities quickly. For Calc II sequence questions, check the direction of the inequality carefully, because the proof or classification can change depending on which side is bounded.
A sequence is bounded below if there is a real number that is less than or equal to every term.
The lower bound does not have to be unique, and any smaller number can also work as a lower bound.
Being bounded below does not automatically mean the sequence converges.
In Calculus II, bounded below often pairs with monotonicity when you study convergence of sequences.
If a sequence keeps dropping without a real floor, it is not bounded below.
A sequence is bounded below if all of its terms stay above some real number. That number is called a lower bound. In Calculus II, this is one of the first properties you check when studying the behavior of a sequence.
Yes. If one number is below every term, then any smaller number is also a lower bound. For example, if a sequence is always at least 2, then 2, 1, 0, and -50 all work.
No. A sequence can stay above a floor and still fail to approach a single limit. In Calc II, bounded below becomes more useful when it is paired with monotonic behavior, especially for decreasing sequences.
Look for a number that every term stays above. For example, 1/n is bounded below by 0 because every term is positive. But -n is not bounded below because the terms keep getting smaller with no real floor.