Combination Theorem for Sequences/Combined Sum Rule

Theorem

Let $X$ be one of the standard number fields $\Q, \R, \C$.

Let $\left \langle {x_n} \right \rangle$ and $\left \langle {y_n} \right \rangle$ be sequences in $X$.

Let $\left \langle {x_n} \right \rangle$ and $\left \langle {y_n} \right \rangle$ be convergent to the following limits:

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

Let $\lambda, \mu \in X$.

Then:

$\displaystyle\lim_{n \to \infty} \left({\lambda x_n + \mu y_n}\right) = \lambda l + \mu m$

Proof

From the Multiple Rule, we have:

• $\displaystyle\lim_{n \to \infty} \left({\lambda x_n}\right) = \lambda l$
• $\displaystyle\lim_{n \to \infty} \left({\mu y_n}\right) = \mu m$

The result now follows directly from the Sum Rule:

$\displaystyle\lim_{n \to \infty} \left({\lambda x_n + \mu y_n}\right) = \lambda l + \mu m$

$\blacksquare$

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 4.8 \ \text{(i)}$