Why doesn't this converge uniformly?

Question

$g_n(x) = \frac{\ln(1+x/n)}{n}$ on $\mathbb{R}$. Don't they all converge to 0?

Answer 1

Please note that$$g_n(ne^n)=\frac{\ln(1+e^n)}n>\frac{\ln(e^n)}n=1.$$Therefore, your sequence cannot converge uniformly to $0$. This, in spite of your (correct) statement that for each individual $x$, $\lim_{n\to\infty}g_n(x)=0$. But uniform convergence is not about that; that's the realm of pointwise convvergence.

Answer 2

Sure (though the domain can't be all of $\mathbb{R}$, presumably you intend $[0,\infty)$ or maybe $(-n,\infty)$). But for any fixed $n$, the whole thing still goes to infinity as $x$ goes to infinity, so $\| g_n - g \|_\infty=+\infty \not \to 0$.

On the other hand the convergence is uniform on compact subsets, which is generally the best that one should hope for on unbounded domains.