Prove that the sequence does not converge uniformly

real-analysis

I want to prove by definition that the following sequence of functions

$${f}_{n}= {1 \over 1+n^2x^2}$$

$x\in [0, 1]$, $n\in \mathbb{N}$, converges uniformly in the interval $(0, 1]$

We can find the pointwise limit $\lim \limits _{n\to \infty} f_n(x_{o})=\lim \limits _{n\to \infty} {1 \over 1+n^2x_{0}^2}$

Consider the following cases

  1. If $x_{0}=1$ $\rightarrow$ $\lim \limits _{n\to \infty} f_n(x_{o})=\lim \limits _{n\to \infty} {1 \over 1+n^2x_{0}^2}=0$

  2. If $0<x_{0}<1$ $\rightarrow$ $\lim \limits _{n\to \infty} f_n(x_{o})=\lim \limits _{n\to \infty} {1 \over 1+n^2x_{0}^2}=0$

We can see that the pointwise limit is $\lim \limits _{n\to \infty} f_n(x)=f(x)=0$ for $x\in (0, 1]$

Now, seeing that the convergence is not uniform has been difficult for me.

My idea is that as the space of bounded real functions $(\mathbf{B}_{\mathbb{R}}((0, 1]), ||.||_{\infty})$, from where $||f||_{\infty}=\sup_{x \in (0, 1]} \{|f_{n}(x)| \}$, is complete then the limit function $f(x)$ should belong to $\mathbf{B}_{\mathbb{R}}((0, 1])$
and reach some kind of contradiction, but I don't see it. Any suggestions?

In addition, I also want to show that

$$\lim \limits _{n\to \infty} \int_{0} ^{1}f_n(x)=\int_{0} ^{1} \lim \limits _{n\to \infty} f_n(x)$$

But taking into account that the pointwise limit is $0$

It all comes down to showing that

$$\lim \limits _{n\to \infty} \int_{0} ^{1}f_n(x)=0$$

Note that

$$\lim \limits _{n\to \infty} \int_{0} ^{1}f_n(x) \, dx=\lim \limits _{n\to \infty} \int_{0} ^{1}{1 \over 1+n^2x^2} \, dx=\lim \limits _{n\to \infty} \frac{1}{n} \arctan(nx)\Bigg|_{0}^{1}$$

So
$$\lim \limits _{n\to \infty} \int_{0} ^{1}f_n(x)=\arctan \cdot \lim \limits _{n\to \infty} \frac{1}{n}=0 $$

Is this conclusion correct?

Any help would be appreciated!

Best Answer

  • If the convergence was uniform on $(0,1],$ i.e. $$\forall\epsilon>0,\exists N,\forall n\ge N,\forall x\in(0,1],|f_n(x)|\le\varepsilon$$ the sequence $(f_n)$ would be uniformly Cauchy on $(0,1]$ i.e. $$\forall\eta>0,\exists N,\forall p,q\ge N,\forall x\in(0,1],|f_p(x)-f_q(x)|\le\eta$$ hence (by continuity of these functions on $[0,1]$) $$\forall\eta>0,\exists N,\forall p,q\ge N,\forall x\in[0,1],|f_p(x)-f_q(x)|\le\eta$$ i.e. on $[0,1],$ $(f_n)$ would be uniformly Cauchy hence uniformly convergent. But it is not, since the $f_n$'s are continuous, whereas the pointwise limit on $[0,1]$ is not continuous at $0.$

  • A more specific and direct argument is: the uniform convergence on $(0,1]$ would entail $$\exists n\in\Bbb N,\forall x\in(0,1],\frac1{1+n^2x^2}\le\frac12$$ i.e. $$\exists n\in\Bbb N,\forall x\in(0,1],x\ge\frac1n,$$ which is clearly false.

  • As for your conclusion about the integral, it is correct (up to the typo in the last line) but can be obtained more naively: $$0\le\int_0^1f_n(x)dx\le\int_0^{1/n}dx+\frac1{n^2}\int_{1/n}^1\frac{dx}{x^2}=\frac1n+\frac1{n^2}(n-1).$$

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