Sum Rule for Sequence in Normed Vector Space

Theorem

Let $\GF \in \set {\R, \C}$.

Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $\GF$.

Let $\sequence {x_n}_{n \in \N}$ and $\sequence {y_n}_{n \in \N}$ be convergent sequences such that:

$x_n \to x$

and:

$y_n \to y$

Then:

$x_n + y_n \to x + y$

Proof

For each $n \in \N$, we have:

\(\ds \norm {\paren {x_n + y_n} - \paren {x + y} }\)

\(=\)

\(\ds \norm {\paren {x_n - x} + \paren {y_n - y} }\)

\(\ds \)

\(\le\)

\(\ds \norm {x_n - x} + \norm {y_n - y}\)

Norm Axiom $\text N 3$: Triangle Inequality

\(\ds \)

\(\to\)

\(\ds 0\)

So from Sequence in Normed Vector Space Convergent to Limit iff Norm of Sequence minus Limit is Null Sequence, we have:

$x_n + y_n \to x + y$

$\blacksquare$