Absolute Value and Maximum Limit Laws

Why These Limit Laws Matter

Sequence limit laws let us find the limit of a more complicated sequence from the limits of simpler ones. Absolute values, maxima, and minima arise naturally when a sequence is defined by cases, when we compare two approximations, or when we measure a distance from zero. Their limit laws can be proved directly from inequalities, so they remain useful even when a sequence changes sign or the larger of two terms changes with \(n\).

Throughout, \(a_n\) and \(b_n\) are real sequences. The statement \(a_n\to A\) means that for every \(\varepsilon>0\), there is an \(N\in\mathbb{N}\) such that \(n\ge N\) implies \(|a_n-A|<\varepsilon\). The same definition applies to the other sequences below.

The Absolute-Value Limit Law

The absolute-value function does not preserve the sign of a number. Consequently, if a sequence converges to a negative number, the limit of its absolute values is positive. The precise law is:

The key estimate is the reverse triangle inequality. For any real numbers \(x\) and \(y\), the triangle inequality gives \(|x|\le |x-y|+|y|\), so \(|x|-|y|\le |x-y|\). Interchanging \(x\) and \(y\) gives \(|y|-|x|\le |x-y|\). Together these show that the absolute values cannot differ by more than the original numbers do:

Apply this with \(x=a_n\) and \(y=A\). If \(|a_n-A|<\varepsilon\), then \(\bigl||a_n|-|A|\bigr|<\varepsilon\). Convergence of \(a_n\) to \(A\) guarantees this inequality for all sufficiently large \(n\), which proves the law.

Maximum and Minimum Limit Laws

Maxima and minima can be expressed using addition, subtraction, and absolute value. For any real \(x\) and \(y\), the identities are

To see why, first suppose \(x\ge y\). Then \(|x-y|=x-y\), so the first expression is \((x+y+x-y)/2=x\), the maximum, and the second is \((x+y-x+y)/2=y\), the minimum. If \(y\ge x\), then \(|x-y|=y-x\); the expressions become \(y\) and \(x\), respectively. The identities therefore hold in either order, including when \(x=y\).

For the maximum, use its identity and the sum and scalar-multiple limit laws. The sequence \(a_n-b_n\) tends to \(A-B\), and the absolute-value law gives \(|a_n-b_n|\to|A-B|\). Therefore

The last equality is the same identity applied to \(A\) and \(B\). For the minimum, the corresponding identity gives

This proof uses only the established limit laws for sums and scalar multiples, together with the absolute-value law. It does not require the same sequence to supply the maximum term at every index.

An Inequality View of the Maximum Law

There is also a useful direct estimate. For any real \(x,y,u,v\), set \(d=\max\{|x-u|,|y-v|\}\). Then \(x\le u+d\) and \(y\le v+d\). Thus \(\max\{x,y\}\le\max\{u,v\}+d\). Reversing the two pairs gives \(\max\{u,v\}\le\max\{x,y\}+d\). Hence

Taking \(x=a_n\), \(y=b_n\), \(u=A\), and \(v=B\) shows directly why convergence of both inputs controls the maximum. Given \(\varepsilon>0\), convergence supplies an index after which both \(|a_n-A|<\varepsilon\) and \(|b_n-B|<\varepsilon\). The displayed estimate then makes the difference between the maxima less than \(\varepsilon\). The identity proof gives the minimum law as well; alternatively, \(\min\{x,y\}=-\max\{-x,-y\}\), so the maximum law and the scalar-multiple law imply it.

Worked Applications

A Common Sign Error

A frequent mistake is to treat absolute value as if it left a negative limit unchanged. For example, if \(a_n=-2+\frac{1}{n}\), then \(a_n\to-2\). The incorrect calculation \(|a_n|\to-2\) fails because absolute values are nonnegative. The correct law gives

Indeed, \(a_n<0\) for every \(n\in\mathbb{N}\), so \(|a_n|=2-\frac{1}{n}\), which tends to \(2\). Always apply the absolute-value law to the limit itself: the result is \(|A|\), not \(A\) unless \(A\ge0\).

Why the Laws Are Useful

These results let us handle sequences whose definitions include distance, error, or a choice between alternatives. The absolute-value law evaluates limits of magnitudes. The maximum and minimum laws let us take limits through a comparison of two convergent sequences, even when their relative order is not fixed. They also support estimates: if both candidate values are close to their limits, the maximum cannot be far from the maximum of those limits.

The hypotheses matter: both input sequences must converge to finite real limits for the stated two-sequence laws to apply. Once they do, the formulas and inequalities establish the answer without requiring a term-by-term case split.