Uniqueness of Limits

Why a Limit Cannot Have Two Values

A sequence may have terms that vary, repeat, or alternate in sign, but if it converges, it approaches one particular real number. The uniqueness of that number is essential: when we calculate a limit in two different ways, the results cannot disagree. It also lets us rule out proposed limits by comparing them with a limit already known.

The theorem itself was established in the tutorial “Convergent Sequences.” We will use it here rather than prove it again. Our focus is on ways to apply uniqueness and on two useful comparison principles: the limit of a difference, and the effect of changing a sequence by an error that tends to zero.

Uniqueness concerns the number a sequence converges to, not the number of times a value occurs among its terms. If a sequence converges and takes the same value infinitely often, that value must be its limit. For instance, terms that alternate between a fixed value and a small error may approach zero even though the fixed value occurs repeatedly only if that value itself is zero. The key question is always whether the distance from the terms to a proposed limit eventually becomes smaller than every positive tolerance.

A common way to use uniqueness is to produce two valid descriptions of a limit. If a sequence is shown to converge to \(L\) by one argument and to \(M\) by another, the theorem immediately gives \(L=M\). A second useful strategy is to compare two sequences whose terms are close. If their difference tends to zero, then their limits, when they exist, must coincide. We prove the comparison facts needed for that strategy next.

The Limit of a Difference

Suppose \(a_n\to L\) and \(b_n\to M\). For each index, the difference \(a_n-b_n\) should approach \(L-M\). To verify this from the definition, rewrite the error as the difference of the two individual errors:

The Triangle Inequality then bounds the error by \(|a_n-L|+|b_n-M|\). Each of these two quantities can be made small eventually. The proof combines the two required thresholds.

Proof. Let \(\varepsilon>0\). Since \(a_n\to L\), there is an \(N_1\in\mathbb{N}_0\) such that \(|a_n-L|<\varepsilon/2\) for every \(n\geq N_1\). Since \(b_n\to M\), there is an \(N_2\in\mathbb{N}_0\) such that \(|b_n-M|<\varepsilon/2\) for every \(n\geq N_2\). Choose \(N\) to be the larger of \(N_1\) and \(N_2\). Then for every \(n\geq N\), both estimates hold. Using the Triangle Inequality,

This is the epsilon-N definition of \(a_n-b_n\to L-M\). \(\square\)

This result turns a comparison of two sequences into a statement about their limits. In particular, if the difference between their terms tends to zero, then the difference between their limits must be zero. The uniqueness theorem lets us express that conclusion as equality of the limits.

Proof. By the Limit of a Difference theorem, \(a_n-b_n\to L-M\). By assumption, the same sequence \(a_n-b_n\) also converges to \(0\). The Uniqueness of Limits theorem therefore gives \(L-M=0\), and hence \(L=M\). \(\square\)

Changing Terms by a Vanishing Error

The preceding comparison has a useful general form. Suppose two sequences differ by an error that tends to zero. If one sequence converges to \(L\), then the other also converges to \(L\). This is stronger than merely saying that two already-convergent sequences have equal limits: it also proves convergence of the second sequence.

Proof. Let \(\varepsilon>0\). Since \(a_n\to L\), choose \(N_1\in\mathbb{N}_0\) such that \(|a_n-L|<\varepsilon/2\) whenever \(n\geq N_1\). Since \(b_n-a_n\to0\), choose \(N_2\in\mathbb{N}_0\) such that \(|b_n-a_n|<\varepsilon/2\) whenever \(n\geq N_2\). Let \(N\) be the larger of \(N_1\) and \(N_2\). For every \(n\geq N\), the Triangle Inequality gives

Therefore \(b_n\to L\), as required. \(\square\)

The theorem makes precise the idea that errors which become arbitrarily small do not change a limit. The error need not have a fixed sign, and it need not be zero at any particular index. What matters is that its absolute value eventually falls below every positive tolerance.

Eventual Agreement and Other Comparisons

A particularly simple vanishing difference occurs when two sequences eventually agree. If there is an index \(K\) such that \(a_n=b_n\) for every \(n\geq K\), then \(b_n-a_n=0\) for every \(n\geq K\), and hence \(b_n-a_n\to0\). The Vanishing Perturbation theorem shows that if one sequence converges to \(L\), the other does too, with the same limit.

This conclusion is consistent with the earlier theorem “Finite Changes Preserve Convergence.” The comparison argument gives another way to see why that fact holds: a difference that is eventually zero tends to zero, so it cannot change the limit. Uniqueness then guarantees that any valid calculation of the limit produces the same answer.

There is an important distinction between “the limits are equal” and “the sequences are equal.” For example, two sequences may differ at every index and still have the same limit, as in the oscillating-error example. Conversely, a small difference for a few initial indices does not establish that two limits agree; the difference must tend to zero, or another argument must establish convergence and identify the limits.

The comparison results also provide a practical check on a proposed answer. If a sequence is known to converge to \(L\), and a second formula differs from it by a term tending to zero, then the second formula must converge to \(L\). If an independent argument claims a different limit for that same sequence, the claims conflict with the Uniqueness of Limits theorem; at least one argument must contain an error.

The main pitfall is to treat closeness at a few indices as though it determines a limit. The relevant condition is eventual control at every positive tolerance. Once the difference tends to zero, however, the comparison is decisive: the two sequences have the same limiting value whenever either one is known to converge.