Let $$g_n(x)=\left\{\begin{array}{ll}nx&\mbox{if }\,0\leq x\leq 1/n\\ 1/(nx)&\mbox{if }\,1/n<x\end{array}\right.$$
my problem is proving that $g_n$ converges punctually to zero, and uniformly only when $x>0$, I'm so slow with the sequences that have some function of $n$ in the domain, someone can help me, I've tried using archmiden property but the case $0<x<1$ is not clear for me... some hint or solution? Thanks in advance.
Hint. If $x>0$ then there is an integer $N$ such that $1/N<x$. Then for $n>N$, $1/n<1/N<x$ and
$$0<g_n(x)=\frac{1}{nx}<\frac{1}{nN}\to 0$$ as $n$ goes to infinity.
P.S. Note that $$\sup_{[0,+\infty)}|g_n(x)-g(x)|=g(1/n)=1$$ where $g=0$ is the pointwise limit. Hence $g_n$ does not converge unformly to $g$ in $[0,+\infty)$.