Real Function Limits:Sequential Criterion

We will now look at a very important theorem known as The Sequential Criterion for a Limit which merges the concept of the limit of a function $f$ at a cluster point $c$ from $A$ with regards to sequences $(a_{n})$ from $A$ that converge to $c$ .

Theorem 1 (The Sequential Criterion for a Limit of a Function): Let $f:A\to \mathbb {R}$ be a function and let $c$ be a cluster point of $A$ . Then $\lim _{x\to c}f(x)=L$ if and only if for all sequences $(a_{n})$ from the domain $A$ where $a_{n}\neq c$ $\forall n\in \mathbb {N}$ and $\lim _{n\to \infty }a_{n}=c$ then $\lim _{n\to \infty }f(a_{n})=L$ .

Consider a function $f$ that has a limit $L$ when $x$ is close to $c$ . Now consider all sequences $(a_{n})$ from the domain $A$ where these sequences converge to $c$ , that is $\lim _{n\to \infty }a_{n}=c$ . The Sequential Criterion for a Limit of a Function says that then that as $n$ goes to infinity, the function $f$ evaluated at these $a_{n}$ will have its limit go to $L$ .

For example, consider the function $f:\mathbb {R} \to \mathbb {R}$ defined by the equation $f(x)=x$ , and suppose we wanted to compute $\lim _{x\to 0}x$ . We should already know that this limit is zero, that is $\lim _{x\to 0}x=0$ . Now consider the sequence $(a_{n})=\left({\frac {1}{n}}\right)$ . This sequence $(a_{n})$ is clearly contained in the domain of $f$ . Furthermore, this sequence converges to 0, that is $\lim _{n\to \infty }{\frac {1}{n}}=0$ . If all such sequences $(a_{n})$ that converge to $0$ have the property that $(f(a_{n}))$ converges to $f(0)=0$ , then we can say that $\lim _{n\to 0}f(x)=0$ .

We will now look at the proof of The Sequential Criterion for a Limit of a Function.

Proof: $\Rightarrow$ Suppose that $\lim _{x\to c}f(x)=L$ , and let $(a_{n})$ be a sequence in $A$ such that $a_{n}\neq c$ $\forall n\in \mathbb {N}$ such that $\lim _{n\to \infty }a_{n}=c$ . We thus want to show that $\lim _{n\to \infty }f(a_{n})=L$ .

Let $\varepsilon >0$ . We are given that $\lim _{x\to c}f(x)=L$ and so for $\varepsilon >0$ there exists a $\delta >0$ such that if $x\in A$ and $0<\mid x-c\mid <\delta$ then we have that $\mid f(x)-L\mid <\varepsilon$ . Now since $\delta >0$ , since we have that $\lim _{n\to \infty }a_{n}=c$ then there exists an $N\in \mathbb {N}$ such that if $n\geq N$ then $\mid a_{n}-c\mid <\delta$ . Therefore $a_{n}\in V_{\delta }(c)\cap A$ .

Therefore it must be that $\mid f(a_{n})-L\mid <\varepsilon$ , in other words, $\forall n\geq N$ we have that $\mid f(a_{n})-L\mid <\varepsilon$ and so $\lim _{n\to \infty }f(a_{n})=L$ .

$\Leftarrow$ Suppose that for all $(a_{n})$ in $A$ such that $a_{n}\neq c$ $\forall n\in \mathbb {N}$ and $\lim _{n\to \infty }a_{n}=c$ , we have that $\lim _{n\to \infty }f(a_{n})=L$ . We want to show that $\lim _{x\to c}f(x)=L$ .

Suppose not, in other words, suppose that $\exists \varepsilon _{0}>0$ such that $\forall \delta >0$ then $\exists x_{\delta }\in A\cap V_{\delta }(c)\setminus \{c\}$ such that $\mid f(x_{\delta })-L\mid \geq \varepsilon _{0}$ . Let $\delta _{n}={\frac {1}{n}}$ . Then there exists $x_{\delta _{n}}=a_{n}\in A\cap V_{\delta _{n}}(c)\setminus \{c\}$ , in other words, $0<\mid a_{n}-c\mid <{\frac {1}{n}}$ and $\lim _{n\to \infty }a_{n}=c$ . However, $\mid f(a_{n})-L\mid \geq \varepsilon _{0}$ so $\lim _{n\to \infty }f(a_{n})\neq L$ , a contradiction. Therefore $\lim _{x\to c}f(x)=L$ .