Let ${f_n}$ be a sequence of functions $f_n:(0,1)\to\Bbb R$ defined by $f_n(x)=1/(nx)$. Prove that ${f_n}$ does not converge uniformly to the zero function.
If you could walk me through this to understand it, that would be appreciated.
real-analysisuniform-convergence
Let ${f_n}$ be a sequence of functions $f_n:(0,1)\to\Bbb R$ defined by $f_n(x)=1/(nx)$. Prove that ${f_n}$ does not converge uniformly to the zero function.
If you could walk me through this to understand it, that would be appreciated.
Best Answer
Or, another way to see this, is to find a sequence of points for wich the function doesn't converge to $0$:
$\forall n\in\mathbb{N}^*, f_n(\frac{1}{n}) = 1$.
And this imply the contradiction of the uniform convergence:
$$\forall N, \exists n > N, \exists x_n=\frac{1}{n},\quad |f_n(x_n)-0|\geq 1$$