Let f(x)= x, g(x)=x.

Question

Let $f(x)=\sin x$, $g(x)=x$.
Statement 1: $f(x)\le g(x)$ for $x$ in $(0,\infty)$.
Statement 2: $f(x)\le1$ for $x$ in $(0,\infty)$ but $g(x)	o\infty$ as $x	o\infty$.

Accepted Answer

Statement 1 ($\sin x\le x$ for $x>0$) is a standard, true inequality. Statement 2 is also true ($\sin x \le 1$ always, and $x	o\infty$), but it does not, by itself, logically establish Statement 1 (boundedness of $\sin x$ combined with unboundedness of $x$ doesn't directly prove the pointwise inequality without further argument). Hence both are true, but Statement 2 is not a correct explanation of Statement 1.

Solutions · 1