Sequential Characterization of Limit at Positive Infinity of Real Function
Theorem
Let $f : \R \to \R$ be a real function.
Let $L$ be a real number.
Then:
- $\ds \lim_{x \to \infty} \map f x = L$
- for all real sequences $\sequence {x_n}_{n \mathop \in \N}$ with $x_n \to \infty$ we have $\map f {x_n} \to L$
where:
- $\ds \lim_{x \mathop \to \infty} \map f x$ denotes the limit at $+\infty$ of $f$.
Corollary
- $\ds \lim_{x \to \infty} \map f x = L$
- for all increasing real sequences $\sequence {x_n}_{n \mathop \in \N}$ with $x_n \to \infty$ we have $\map f {x_n} \to L$
where:
- $\ds \lim_{x \mathop \to \infty} \map f x$ denotes the limit at $+\infty$ of $f$.
Proof
Necessary Condition
Suppose that:
- $\ds \lim_{x \to \infty} \map f x = L$
Let $\sequence {x_n}_{n \mathop \in \N}$ be a real sequence with $x_n \to \infty$.
Let $\epsilon > 0$.
From the definition of limit at infinity, we have:
- there exists $M > 0$ such that for all $x > M$ we have $\size {\map f x - L} < \epsilon$.
Since $\sequence {x_n}_{n \mathop \in \N}$ diverges to $+\infty$, we have:
- there exists $N \in \N$ such that $x_n > M$ for all $n \ge N$.
That is, for all $n \ge N$, we have:
- $\size {\map f {x_n} - L} < \epsilon$
Since $\epsilon$ was arbitrary, we have:
- $\map f {x_n} \to L$
Since $\sequence {x_n}_{n \mathop \in \N}$ was arbitrary:
- for all real sequences $\sequence {x_n}_{n \mathop \in \N}$ with $x_n \to \infty$ we have $\map f {x_n} \to L$.
So, if:
- $\ds \lim_{x \to \infty} \map f x = L$
then:
- for all real sequences $\sequence {x_n}_{n \mathop \in \N}$ with $x_n \to \infty$ we have $\map f {x_n} \to L$.
$\Box$
Sufficient Condition
We prove the contrapositive.
Suppose that it is not the case that:
- $\ds \lim_{n \mathop \to \infty} \map f x = L$
We show that:
- there exists a real sequence $\sequence {x_n}_{n \mathop \in \N}$ with $x_n \to \infty$ but such that $\sequence {\map f {x_n} }_{n \mathop \in \N}$ does not converge to $L$.
Then:
- there exists some $\epsilon > 0$ such that for all $M > 0$ there exists $x > M$ such that $\size {\map f x - L} \ge \epsilon$.
We construct $\sequence {x_n}_{n \mathop \in \N}$ inductively.
Pick $x_1$ such that $x_1 > 1$ and:
- $\size {\map f {x_1} - L} \ge \epsilon$
Given $x_1, x_2, \ldots, x_{n - 1}$, pick $x_n$ so that:
- $x_n > \max \set {n, x_1, x_2, \ldots, x_{n - 1} } \ge x_{n - 1}$
and:
- $\size {\map f {x_n} - L} \ge \epsilon$
Then $\sequence {x_n}_{n \mathop \in \N}$ is increasing.
We also have:
- $x_n > n$ for each $n$
so $\sequence {x_n}_{n \mathop \in \N}$ is unbounded above.
From Unbounded Monotone Sequence Diverges to Infinity: Increasing, we therefore have:
- $x_n \to \infty$
Since:
- $\size {\map f {x_n} - L} \ge \epsilon$
there exists no $N \in \N$ such that:
- $\size {\map f {x_n} - L} < \epsilon$
for each $n \ge N$.
So:
- $\sequence {\map f {x_n} }_{n \mathop \in \N}$ does not converge to $L$.
We conclude that if it is not the case that:
- $\ds \lim_{x \mathop \to \infty} \map f x = L$
then:
- there exists a real sequence $\sequence {x_n}_{n \mathop \in \N}$ with $x_n \to \infty$ but such that $\sequence {\map f {x_n} }_{n \mathop \in \N}$ does not converge to $L$.
So, if:
- $\ds \lim_{x \mathop \to \infty} \map f x = L$
then:
- for all real sequences $\sequence {x_n}_{n \mathop \in \N}$ with $x_n \to \infty$ we have $\map f {x_n} \to L$.
$\blacksquare$