Algebraic Limit Theorem in Vector Spaces

Refer Rudin p46, 3.4.

The Algebraic Limit Theorem, formulated in the real field, can be extended to vector spaces like ${\mathbb{R}}^n$.

A sequence of vectors is a function $f:\mathbb{N}\to {\mathbb{R}}^n$, with $f(n)$ being the $n$th term of the sequence.

First, we need to establish what it means for a sequence of vectors to converge.

Slot-wise convergence

Theorem 1(Theorem).

Suppose ${\mathbf{x}}_n\in {\mathbb{R}}^k$ and ${\mathbf{x}}_n=({\alpha }_{1,n},{\alpha }_{2,n},\dots ,{\alpha }_{k,n})$.
Then $({\mathbf{x}}_n)\to \mathbf{x}=({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_k)$ if and only if

$$\underset{n\to \infty }{\lim }{\alpha }_{j,n}={\alpha }_j\forall 1\le j\le k.$$

Proof
Forward direction:
Since $({\mathbf{x}}_n)\to \mathbf{x}$, we have

$$\begin{array}{rl}& |{\mathbf{x}}_n-\mathbf{x}|<ϵ\\ ⟹& \sqrt{\sum _{j=1}^k({\alpha }_{j,n}-{\alpha }_j{)}^2}<ϵ\\ ⟹& |{\alpha }_{j,n}-{\alpha }_j|<ϵ\forall 1\le j\le k\end{array}$$

for all $n\ge N$ for some $N$. Thus, ${\lim }_{n\to \infty }{\alpha }_{j,n}={\alpha }_j$ for all $1\le j\le k$.

Backward direction:
Let $ϵ>0$ be arbitrary. Choose $N_1,N_2,\dots ,N_k$ such that

$$\begin{array}{r}|{\alpha }_{j,n}-{\alpha }_j|<\frac{ϵ}{\sqrt{k}}\end{array}$$

For all $n\ge N_j$ respectively. Let $N=\max \{N_1,N_2,\dots ,N_k\}$. Squaring and adding the $k$ inequalities, we get

$$\begin{array}{r}\sum _{j=1}^k({\alpha }_{j,n}-{\alpha }_j{)}^2<{ϵ}^2\end{array}$$

for all $n\ge N$. ❏

Note that while this proof is done for vectors in ${\mathbb{R}}^k$, it also works for ${\mathbb{C}}^k$.

Algebra of limits of vector sequences

Now, we are ready to perform algebra on limits of vector sequences.

Theorem 2(Theorem).

Let $({\mathbf{x}}_n)$ and $({\mathbf{y}}_n)$ be sequences in ${\mathbb{R}}^k$, $({\beta }_n)$ be a sequence in $\mathbb{R}$, and $({\mathbf{x}}_n)\to \mathbf{x}$, $({\mathbf{y}}_n)\to \mathbf{y}$, $({\beta }_n)\to \beta$. Then,

1. ${\lim }_{n\to \infty }({\mathbf{x}}_n+{\mathbf{y}}_n)=\mathbf{x}+\mathbf{y}$
2. ${\lim }_{n\to \infty }{\mathbf{x}}_n\cdot {\mathbf{y}}_n=\mathbf{x}\cdot \mathbf{y}$
3. ${\lim }_{n\to \infty }{\beta }_n{\mathbf{x}}_n=\beta \mathbf{x}$

All of these follow from the previous theorem and the Algebraic Limit Theorem in $\mathbb{R}$.