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

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Corollary

Let $f : \R \to \R$ be a real function.

Let $L$ be a real number.


Then:

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

if and only if:

for all decreasing 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:

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

Then, from Sequential Characterization of Limit at Minus 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 decreasing 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

Suppose that:

for all decreasing 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 Negative Infinity Diverges to Negative 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:

$\map f {x_n} \to L$

$\blacksquare$