I want to show if the following sequence of functions converges pointwise or uniformly on $R.$ $f_n(x) = 0$ if $x<n $ and $1/x$ if $x\geq n $. Here's what I did, fix $x\in R $. for all $n \geq 1/x$, $f_n(x) \rightarrow1/x$. Furthermore, $f_n(0)=0$. So $f_n$ converges to $f$ s.t. $f(x) =0$ if $x =0$ and $1/x$ if $0<x\leq 1$. So it's pointwise convergent. Is this correct?
For uniform convergent I say take $x$ s.t. $0<x<n$ then $f_n(x)-f(x) = 0-1/x $ which doesn't $\rightarrow 0 $. So it's not uniformly convergent.
Observe that: $$0\leq f_n(x)\leq\frac1n$$ for every $x\in\mathbb R$.
This together with $\lim_{n\to\infty}\frac1n=0$ tells us that $f_n$ converges uniformly (hence also pointwise) to the zero function.