I have the following exercise from my book:

For each $n \in \mathbb{N}$ and each $x\in (-1, 1),$ define

$$f_{n}(x) = \sqrt{x^{2} + \frac{1}{n}}$$

and define $f(x) = |x|$. Prove that the sequence $\{f_{n}\}$ converges uniformly on the open interval $(-1, 1)$ to the function $f$. Check that each function $f_{n}$ is continuously differentiable, whereas the limit function $f$ is not differentiable at $x = 0$.

My attempt:

First, we need to show $\forall \epsilon > 0$, there exists an index $N$ such that

$$|f(x) - f_{n}(x)| < \epsilon $$

for all $n \geq N$ and all $x \in D$. So, we have

$$\left||x| - \sqrt{x^2 + 1/n}\right|.$$

But since $x^{2} + 1/n > 0$ always, we have

$$\left||x| - \sqrt{x^2 + 1/n}\right| = \left||x| - |\sqrt{x^{2} + 1/n}|\right| \leq \left|x - \sqrt{x^{2} + 1/n}\right|,$$

where the last equality follows from the reverse triangle inequality. Then,

$$\left|x - \sqrt{x^{2} + 1/n}\right| \leq |x| < \epsilon.$$

So, choose $N = \lceil{\epsilon}\rceil + 1$. Does this choice of $N$ work? I don't really know. Can someone help me with the rest of the problem?

  • 1
    $\begingroup$ Why is $|x|<\epsilon$? Your choice of $\epsilon$ can't depend on $x$. Also, if $x<0$, then $|x-\sqrt{x^2+1/n}|>|x|.$ My professor taught me a cool method to solve this problem using the squeeze theorem for uniform convergence. Can you show $|x|\leq\sqrt{x^2+1/n}\leq(|x|+1/n)?$ $\endgroup$ – Melody Dec 10 '18 at 1:50
  • $\begingroup$ I cannot prove both inequalities. Can you help me? $\endgroup$ – joseph Dec 10 '18 at 1:54
  • $\begingroup$ Consider $|x|^2\leq|x|^2+1/n\leq|x|^2+2|x|/n+1/n^2$. Now taking square roots gives us the inequality I mentioned earlier. $\endgroup$ – Melody Dec 10 '18 at 1:59

Observe that $$\left(\vert x\vert +\frac{1}{\sqrt{n}}\right)^2=x^2+\frac{1}{n}+2 \vert x \vert \frac{1}{\sqrt{n}}=f_n^2+2 \vert x \vert \frac{1}{\sqrt{n}} \geq f_n^2 \geq 0$$ so $$0 \leq f_n -\vert x \vert\leq \frac{1}{\sqrt{n}} \longrightarrow 0$$

  • $\begingroup$ But doesn't this show pointwise convergence and not uniform convergence? $\endgroup$ – joseph Dec 10 '18 at 1:55
  • $\begingroup$ it's uniform because it only depends on n $\endgroup$ – user29418 Dec 10 '18 at 1:56
  • $\begingroup$ @Joseph: the convergence is uniform! $\endgroup$ – Chinnapparaj R Dec 10 '18 at 1:57
  • $\begingroup$ how'd you get the expression after "so"? $\endgroup$ – joseph Dec 10 '18 at 2:04
  • 1
    $\begingroup$ computing $f_n'$ is easy and observe each $f_n$ is differentiable and $f_n'$ is continuous, so $f_n$ is continuously differentiable. $\endgroup$ – Chinnapparaj R Dec 10 '18 at 2:24

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