In a nutshell, the algebraic limit theorem and order limit theorem are very similar in that they both describe limits for bounded convergent sequences. While the algebraic theorem uses +, –, *, and ÷, the order theorem uses ≤ and ≥.
The algebraic limit theorem shows us that sequences behave in an orderly fashion when subjected to mathematical operations.
Formally stated, the algebraic limit theorem is :
If a = lim an and b = lim bn, then
- lim(can) = ca for all c ∈ ℝ,
- lim(an + bn) = a + b,
- lim(anbn) = ab,
- lim(an/bn) = a/b provided b ≠ 0.
Each of these statements has their own proof. For example, the proof for the first statement is as follows:
Proof : Assuming that c ≠ 0, the goal is to show the sequence converges to c · a.
- Let ε = an arbitrary positive number. We want to find the point in the sequence where |can – ca| < ε.
- |can – ca| = |c| |an – a|
- Choose an N so that:
This is true whenever n ≥ N. Here, we’re making N very, very small.
- Show that N works. For all n ≥ N:
Algebraic Limit Theorem Example: A Worked Proof 
Example question: Show that If (xn) → 2, then ((2xn – 1)/3) → 1.
- Start with the given information: Xn → 2.
- Rewrite as a pair of fractions:
- Substitute the values into the algebraic limit theorem, which tells us that can → ca. This results (with a little numerical manipulation) in:
- We know by the Algebraic Limit Theorem that an + bn → a + b., so:
Alternative Definition for Functions
- lim (f(x) + g(x)) = lim f(x) + lim g(x)
- lim (f(x) – g(x)) = lim f(x) – lim g(x)
- lim (f(x) · g(x)) = lim f(x) · lim g(x)
- lim (f(x) / g(x)) = lim f(x) / lim g(x)
The indeterminate limits (on the left) may exist even when the limits on the right do not. If any of the theorems don’t give you a limit, try an alternate method like the sandwich theorem or L’Hospital’s rule.
Suppose a = lim an and b = lim bn. The order limit theorem states that :
- If an ≥ 0 for all n ∈ ℕ, then a ≥ 0.
- If an ≤ bn for all n ∈ ℕ, then a ≤ b.
- Assuming there is a c ∈ ℝ:
- If c ≤ bn for all n ∈ ℕ, then c ≤ b.
- If an ≤ c for all n ∈ ℕ, then a ≤ c.
 Bakker, L. Math 341 Lecture #8. §2.3: The Algebraic and Order Limit Theorems. Retrieved August 19, 2021 from: https://math.byu.edu/~bakker/M341/Lectures/Lec08.pdf
 Chapter 2: Sequence and Series. Retrieved August 19, 2021 from: http://econdse.org/wp-content/uploads/2018/08/sequence-reading.pdf
 Lytle, B. Introduction to the convergence of sequences. Retrieved August 19, 2021 from: https://math.uchicago.edu/~may/REU2015/REUPapers/Lytle.pdf