If $\displaystyle y_n(x)=nxe^{-nx^2}$ show that for every $\displaystyle x \in [0,1]$, $\displaystyle y_n\rightarrow 0$. I can show this for x=0 and x=1 but not sure for $\displaystyle x \in (0,1)$. Help much appreciated.
If $\displaystyle y_n(x)=nxe^{-nx^2}$ show that for every $\displaystyle x \in [0,1]$, $\displaystyle y_n\rightarrow 0$. I can show this for x=0 and x=1 but not sure for $\displaystyle x \in (0,1)$. Help much appreciated.
You know that for all $\displaystyle x$ we have $\displaystyle e^{-nx}>0~\&~n>0$.
If $\displaystyle 0<x<1$ then $\displaystyle 0<nxe^{-nx}<ne^{-nx}.$