# Combination Theorem for Sequences/Real/Product Rule

## Theorem

Let $\sequence {x_n}$ and $\sequence {y_n}$ be sequences in $\R$.

Let $\sequence {x_n}$ and $\sequence {y_n}$ be convergent to the following limits:

- $\displaystyle \lim_{n \mathop \to \infty} x_n = l$
- $\displaystyle \lim_{n \mathop \to \infty} y_n = m$

Then:

- $\displaystyle \lim_{n \mathop \to \infty} \paren {x_n y_n} = l m$

## Proof

Because $\sequence {x_n}$ converges, it is bounded by Convergent Sequence is Bounded.

Suppose $\size {x_n} \le K$ for $n = 1, 2, 3, \ldots$.

Then:

\(\displaystyle \size {x_n y_n - l m}\) | \(=\) | \(\displaystyle \size {x_n y_n - x_n m + x_n m - l m}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(\le\) | \(\displaystyle \size {x_n y_n - x_n m} + \size {x_n m - l m}\) | $\quad$ Triangle Inequality for Real Numbers | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \size {x_n} \size {y_n - m} + \size m \size {x_n - l}\) | $\quad$ Absolute Value of Product | $\quad$ | |||||||||

\(\displaystyle \) | \(\le\) | \(\displaystyle K \size {y_n - m} + \size m \size {x_n - l}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=:\) | \(\displaystyle z_n\) | $\quad$ | $\quad$ |

But $x_n \to l$ as $n \to \infty$.

So $\size {x_n - l} \to 0$ as $n \to \infty$ from Convergent Sequence Minus Limit.

Similarly $\size {y_n - m} \to 0$ as $n \to \infty$.

From the Combined Sum Rule for Real Sequences:

- $\displaystyle \lim_{n \mathop \to \infty} \paren {\lambda x_n + \mu y_n} = \lambda l + \mu m$, $z_n \to 0$ as $n \to \infty$

The result follows by the Squeeze Theorem for Sequences of Complex Numbers (which applies as well to real as to complex sequences).

$\blacksquare$

## Sources

- 1953: Walter Rudin:
*Principles of Mathematical Analysis*: $3.3c$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $\S 1.2$: Real Sequences: Proposition $1.2.11 \ \text {(b)}$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 4.8 \ \text {(ii)}$: Criteria for convergence - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): Appendix: $\S 18.3$: Combination theorem - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products: Exercise $5$