A continuous function $f:\mathbb{R}^n\to \mathbb{R}$ is called coercive if $$\lim_{\|x\|\to\infty} f(x)=\infty.$$
I'm confused as to how to interpret this limit. I have the 2 following possible interpretations, which I don't think are equivalent.
- For each $N\in \mathbb{R}$, there is some $M>0$ such that $\|x\|>M\implies f(x)>N$.
- For each (not necessarily continous) path $p:\mathbb{R}\to \mathbb{R}^n$, where $$\lim_{t\to\infty} \|p(t)\|=\infty,$$ we have that $$\lim_{t\to\infty}f(p(t))=\infty.$$
I believe it to be the 2nd statement, but the textbook I'm reading seems to use the 1st statement. Can anyone tell me which one is correct or if they're equivalent? Thanks.
There are uncountably many choices for $p$, so we can't necessarily compute the $M$-value for each of them, as the collection of $M$'s might not be bounded.
Solution
$(1)\implies (2)$ is obvious.
For $(2)\implies (1)$, assume (1) fails. There take some $N$ such that for each $M$, there is some $p(M)$ such that $||p(M)||>M$ and $f(p(M))\le N$. Then the first limit in (2) holds, but the 2nd limit is $\le N$, so (2) fails.