Combination Theorem for Sequences/Real/Sum Rule
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Contents
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
Let $\epsilon > 0$ be given.
Then $\dfrac \epsilon 2 > 0$.
We are given that:
- $\displaystyle \lim_{n \mathop \to \infty} x_n = l$
- $\displaystyle \lim_{n \mathop \to \infty} y_n = m$
By definition of the limit of a real sequence, we can find $N_1$ such that:
- $\forall n > N_1: \size {x_n - l} < \dfrac \epsilon 2$
where $\size {x_n - l}$ denotes the absolute value of $x_n - l$
Similarly we can find $N_2$ such that:
- $\forall n > N_2: \size {y_n - m} < \dfrac \epsilon 2$
Let $N = \max \set {N_1, N_2}$.
Then if $n > N$, both the above inequalities will be true:
- $n > N_1$
- $n > N_2$
Thus $\forall n > N$:
\(\displaystyle \size {\paren {x_n + y_n} - \paren {l + m} }\) | \(=\) | \(\displaystyle \size {\paren {x_n - l} + \paren {y_n - m} }\) | $\quad$ | $\quad$ | |||||||||
\(\displaystyle \) | \(\le\) | \(\displaystyle \size {x_n - l} + \size {y_n - m}\) | $\quad$ Triangle Inequality for Real Numbers | $\quad$ | |||||||||
\(\displaystyle \) | \(<\) | \(\displaystyle \frac \epsilon 2 + \frac \epsilon 2\) | $\quad$ | $\quad$ | |||||||||
\(\displaystyle \) | \(=\) | \(\displaystyle \epsilon\) | $\quad$ | $\quad$ |
Hence the result:
- $\displaystyle \lim_{n \mathop \to \infty} \paren {x_n + y_n} = l + m$
$\blacksquare$
Also see
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $\S 1.2$: Real Sequences: Proposition $1.2.11 \ \text {(a)}$
- 1953: Walter Rudin: Principles of Mathematical Analysis: $3.3a$