Useful limit theorem

22.03.2012

Here is a useful limit theorem which can be used to prove many of the commonly needed limit theorems as special cases.

Claim

Let $(X^*, T_{X^*})$ , $(X,T_X)$ , $(Y, T_Y)$ , and $(Z, T_Z)$ be topological spaces, where $X$ is a subspace of $X^*$ . Let $f : X \to Y$ be such that $\lim_{x \to p} f(x) = c$ , where $c \in Y$ and $p \in \overline{X}$ . Let $g : Y \to Z$ be continuous at $c$ . Then

$\begin{equation} \begin{split} \lim_{x \to p} g(f(x)) = g(c). \end{split} \end{equation}$

Example 1 : The limit of evaluations is the evaluation of the limit

Let $f : X \to X$ be such that $f(x) = x$ , and $g : X \to Z$ be continuous at $p \in X$ . Then

$\begin{equation} \begin{split} \lim_{x \to p} g(x) = g(p). \end{split} \end{equation}$

Actually, this is equivalent to continuity at $p$ .

Example 2: The limit of sequential evaluations is the evaluation of the limit

Let $f : \mathbb{N} \to Y$ , where $\mathbb{N}$ is a subspace of $\mathbb{N}^* = \mathbb{N} \cup \{\infty\}$ . Let $g:Y \to Z$ be a continuous function. Then

$\begin{equation} \begin{split} \lim_{n \to \infty} g(f(n)) = g\left(\lim_{n \to \infty} f(n)\right). \end{split} \end{equation}$

Example 3 : The limit of a component is the component of a limit

Let $f : X \to Z^n$ be such that $f(x) = (f_1(x), \dots, f_n(x))$ and $\pi_i : Z^n \to Z$ be such that $\pi_i(z) = z_i$ . Since the projection $\pi_i$ is continuous,

$\begin{equation} \begin{split} \lim_{x \to p} f_i(x) = \pi_i\left(\lim_{x \to p} f(x)\right) = c_i. \end{split} \end{equation}$

Actually, the reverse implication also holds: if the limits of all component functions exist, then so does the limit of the function (with the relation above).

Example 4 : Semigroup addition

Let $X$ and $Z$ be topological semigroups (e.g vector spaces), $f : X \to Z^2$ , and $g : Z^2 \to Z$ be such that $g(x,y) = x + y$ . Since addition is continuous,

$\begin{equation} \begin{split} \lim_{x \to p} [f_1(x) + f_2(x)] = \lim_{x \to p} f_1(x) + \lim_{x \to p} f_2(x). \end{split} \end{equation}$

We also used the result of example 3 here.

Example 5 : Module multiplication

Let $X$ , $Y$ , and $Z$ be topological $R$ -modules (e.g. vector spaces), $f : X \to Y$ , and $g : Y \to Z$ be such that $g(y) = \alpha y$ , where $\alpha \in \mathbb{R}$ . Since by definition the multiplication is continuous, so is its restriction to a fixed $\alpha$ , and thus

$\begin{equation} \begin{split} \lim_{x \to p} [\alpha f(x)] = \alpha \lim_{x \to p} f(x). \end{split} \end{equation}$

Proof

Denote by $T_Z(z)$ the neighborhoods of a point $z \in Z$ (and similarly for other spaces). Let $c = \lim_{x \to p} f(x)$ , and $W \in T_Z(g(c))$ . Since $g$ is continuous at $c$ , there exists $V \in T_Y(c)$ such that $g(V) \subset W$ . By the definition of a limit of a function, there exists $U \subset T_{X^*}(p)$ such that $f(X \cap U \setminus \{p\}) \subset V$ . Therefore $g(f(X \cap U \setminus \{p\})) \subset W$ , i.e. $\lim_{x \to p} g(f(x)) = g(c)$ . QED.