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:

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


 * $x_n + y_n \to x + y$