FALSE: $f<g$ near $c$ implies $\lim f<\lim g$

False claim: let $A\subseteq \mathbb{R}$, let $c$ be a limit point of $A$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$), let $f,g:A\to \mathbb{R}$ have limits at $c$ (The $\epsilon$-$\delta$ limit ${\lim }_{x\to c}f(x)=L$ of $f:A\to \mathbb{R}$ at a limit point $c$ of $A$), and suppose there is a real $\eta >0$ with

$f(x)<g(x)\text{for every }x\in A\text{ with }0<\mathrm{\mid }x-c\mathrm{\mid }<\eta \mathrm{.}$

Then ${\lim }_{x\to c}f(x)<{\lim }_{x\to c}g(x)$.

What is true is the non-strict version, If $f\le g$ on a punctured neighbourhood of $c$ then $\lim f\le \lim g$, non-strictly: the hypothesis $f\le g$ near $c$ gives $\lim f\le \lim g$, and that conclusion cannot be improved even when the hypothesis is strengthened to a strict inequality at every point.

Why the strengthening fails. Strictness at each point is not a uniform statement: it says $g(x)-f(x)>0$ for every $x$ near $c$, with no lower bound on that positive quantity. The limit only sees the limit of $g-f$, and a function that is positive everywhere may have limit $0$. What does survive is the uniform version: if $g(x)-f(x)\ge \kappa$ near $c$ for a fixed real $\kappa >0$, then $\lim g-\lim f\ge \kappa >0$, by applying If $f\le g$ on a punctured neighbourhood of $c$ then $\lim f\le \lim g$, non-strictly to $f+\kappa$ and $g$.

• The non-strict conclusion is sharp, and this witness shows it: the hypothesis is as strong as a pointwise strict inequality can be, and the conclusion still degenerates to equality.

• The same phenomenon for sequences is the reason Limits preserve non-strict inequalities is stated non-strictly; the witness there is a positive null sequence compared with the constant $0$, which is the sequential shadow of the pair above.

• A common misuse. From $f<g$ near $c$ one may conclude $\lim f\le \lim g$ and nothing more; in particular one may not conclude that $\lim g>0$ from $g>0$ near $c$. To get a strict conclusion one needs either a uniform gap, as noted above, or a separate argument such as If ${\lim }_{x\to c}f(x)=L\mathrm{\ne }0$ then $\mathrm{\mid }f\mathrm{\mid }>\mathrm{\mid }L\mathrm{\mid }\mathrm{/}2$ on a punctured neighbourhood of $c$; in particular if $L>0$ then $f>L\mathrm{/}2>0$ there, which works in the opposite direction: from a nonzero limit to a bound on the values.