limit and uniform convergence of sequence of function I have a sequence of function $g_n$:
$$ g_n(x) = xe^{-nx}, \qquad \text{for   } x \in [0,\infty )
$$
I need to find the limit,
and determine the uniform convergnece of the sequence.
I think the limit is 0, but having trouble determining whether the convergence is uniform.

$g_n$ tends to a zero function and it converges uniformly. The first is obvious because the exponential term is less than or equal to $1$ for every $x$ and multiplying by $x$ when $x=0$ gives $0$ at $x=0$. For $x>0$ the exponential term tends to $0$ so $g_n\rightarrow 0$ pointwise.
It converges uniformyl because when you take the derivative of the function make it equal to zero. you will get $x=1/n$. When you insert it back you will get $a_n=1/(en)$ and since $a_n\rightarrow 0$ as $n\rightarrow \infty$, the convergence is uniform.
