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 $\map f {x_n} \to L$.

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 infinity, 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:


 * $\map f {x_n} \to L$

Sufficient Condition
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:


 * $\size {\map f x - L} \ge \epsilon$