Question

Let ($\text{a}_\text{n}$) and ($\text{b}_\text{n}$) are two convergent sequences of real numbers, then which of the following is always true ?

• A. ($\text{a}_\text{n}$) + ($\text{b}_\text{n}$) is convergent
• B. $\frac{(\text{a}_\text{n})}{(\text{b}_\text{n})}$ is always divergent
• C. $\frac{(\text{b}_\text{n})}{(\text{a}_\text{n})}$ is always convergent
• D. ($\text{a}_\text{n}$) - ($\text{b}_\text{n}$) is not convergent

Hint:

Algebra of Limits for convergent sequences \( (a_n) \to A \) and \( (b_n) \to B \):
- \( (a_n \pm b_n) \to A \pm B \).
- \( (a_n \cdot b_n) \to A \cdot B \).
- \( (a_n / b_n) \to A / B \) (provided \( B \neq 0 \)).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We evaluate the algebraic limit theorems for convergent sequences of real numbers.

Step 2: Detailed Explanation:
Let \( (a_n) \) and \( (b_n) \) be two convergent sequences such that:
\[ \lim_{n \to \infty} a_n = A \quad \text{and} \quad \lim_{n \to \infty} b_n = B \]
where \( A \) and \( B \) are finite real numbers.
Let us examine each option based on real analysis limit theorems:
- Option A:
According to the additive limit theorem, the sum of two convergent sequences is always convergent, and its limit is the sum of their individual limits:
\[ \lim_{n \to \infty} (a_n + b_n) = \lim_{n \to \infty} a_n + \lim_{n \to \infty} b_n = A + B \]
Thus, Option A is always true.
- Option B & C:
For the quotient of two convergent sequences \( \frac{a_n}{b_n} \) to be convergent, we must have \( B \neq 0 \) and \( b_n \neq 0 \) for all \( n \).
If these conditions are met, then \( \lim_{n\to\infty} \frac{a_n}{b_n} = \frac{A}{B} \), which is convergent, making Option B false.
However, if the denominator sequence converges to 0, the quotient is undefined or divergent, which means the quotient is not always convergent, making Option C false.
- Option D:
Similarly, the difference sequence \( (a_n - b_n) \) is always convergent and converges to \( A - B \), making Option D false.

Step 3: Final Answer:
The statement that is always true is ($\text{a}_\text{n}$) + ($\text{b}_\text{n}$) is convergent (Option A).