Sequential Characterization of Limit at Positive Infinity of Real Function/Corollary

Corollary
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 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 $f$.

Necessary Condition
Suppose that:


 * $\ds \lim_{x \to \infty} \map f x = L$

Then, from Sequential Characterisation of Limit at Positive Infinity of Real Function, we have:


 * 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, in particular:


 * for all increasing real sequences $\sequence {x_n}_{n \mathop \in \N}$ with $x_n \to \infty$ we have $\map f {x_n} \to L$.

Sufficient Condition
Suppose that:


 * for all increasing real sequences $\sequence {x_n}_{n \mathop \in \N}$ with $x_n \to \infty$ we have $\map f {x_n} \to L$.

Let $\sequence {x_n}_{n \mathop \in \N}$ be a real sequence with $x_n \to \infty$.

Let $\sequence {x_{n_j} }_{j \mathop \in \N}$ be a subsequence of $\sequence {x_n}_{n \mathop \in \N}$.

From Peak Point Lemma, there exists a monotone subsequence $\sequence {x_{n_{j_k} } }_{k \mathop \in \N}$ of $\sequence {x_{n_j} }_{j \mathop \in \N}$.

From Subsequence of Real Sequence Diverging to Positive Infinity Diverges to Positive Infinity, we have:


 * $x_{n_{j_k} } \to \infty$

So, from the hypothesis, we have:


 * $\map f {x_{n_{j_k} } } \to L$

So:


 * any subsequence of $\sequence {\map f {x_n} }_{n \mathop \in \N}$ has a subsequence converging to $L$.

So, from Real Sequence with all Subsequences having Convergent Subsequence to Limit Converges to Same Limit, we have:


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