Show that the following sequence is convergent pointwise but not uniformly convergent.

Question

Let the sequence of fuctions $f_n$ such that $f_n:[0,1] \rightarrow R$ where $f_n(x)=nxe^{-nx^2}$. Show that the sequence is convergent pointwise to $f$. Futhermore show that $f$ is R-Integrable and $\int_0^1 f =0$. Show that $$ \lim_{n \rightarrow +\infty}\int_0^1f_n=\frac{1}{2} $$ Finally, show that the convergence is not uniform.

MY ATTEMPT

If $x \in ]0,1]$ $$ \lim_{n\rightarrow +\infty}{nxe^{-nx^2}}=x\lim_{n\rightarrow +\infty}{\frac{n}{e^{nx^2}}}=\frac{1}{x}\lim_{n\rightarrow+\infty}{\frac{1}{e^{nx^2}}}=0 $$ If $x =0$, so $\lim_{n\rightarrow+\infty}{nxe^{-nx^2}}=0$

Therefore, the sequence is convergent pointwise to $f=0$. So $\int_0^1f=0$, $f$ is R-integrable (is a step fuction) and $$ \lim_{n \rightarrow +\infty}\int_0^1 f_n= \frac{1}{2} $$ solving by substitution.

MY ANSWER: How to show that the convergence is not uniform?

In the next exercise, was given the function $$f_n(x)= \frac{n^2x}{1+n^3x^2}$$ He asks to show the convergence is pointwise and show that the convergence is not uniform, the strategy is the same?

Answer 1

Hint: For uniformly convergent sequences of Riemann integrable functions, the Riemann integral behaves like $\lim_{n \to \infty} \int f_n = \int \lim_{n\to \infty} f_n $.

Answer 2

Note: To write $lim_{n \rightarrow \infty}$ $\int f_{n}$ = $\int lim_{n \rightarrow \infty} f_{n} $ you may need one of these conditions first $sup |f_{n}| < K $ for some constant $K$ or second $sup |f_{n}|$ is integrable or third $|f_{n}|<|f_{n+1}|$ provided (for all ) $lim_{n \rightarrow \infty} f_{n}(x) =:f(x) $ exist for all x. Since one of these conditions holds in your homework (check), you can interchange limit and integral.