# Uniform convergence of a sequence of functions

## Homework Statement

This is a translation so sorry in advance if there are funky words in here[/B]
f: ℝ→ℝ a function 2 time differentiable on ℝ. The second derivative f'' is bounded on ℝ.
Show that the sequence on functions $$n[f(x + 1/n) - f(x)]$$ converges uniformly on f'(x) on ℝ.

## Homework Equations

In my attempt I used the definition of differentiation:
$$\lim_{x\to\infty} \frac{f(x+h) - f(x)}{h}$$

and I used a criteria for uniform convergence of sequences of functions:

fn fonverges to f uniformely on A
if and on if
for all ε > 0, ∃ N ∈ ℕ, for which
$$\lim_{x\to\infty} sup|fn - f(x)| \leqslant \varepsilon$$
for all n≥ N, for all x ∈ A

## The Attempt at a Solution

[/B]
I arrived to an answer but I fear I got sidetracked somewhere because I did not use the bounded second derivative.

I rewrote
$$n[f(x + 1/n) - f(x)] = \frac{[f(x + 1/n) - f(x)]}{1/n}$$

Now this looks awefully like the derivative of fn for all x which is:
$$\lim_{n\to\infty} \frac{[f(x + 1/n) - f(x)]}{1/n}$$

And now I applied the definition of the uniform convergence which is:
$$\lim_{n\to\infty} sup| \frac{[f(x + 1/n) - f(x)]}{1/n} - f'(x)| \leqslant \varepsilon$$

And therefore, I proved the uniform convergence to f'(x) on ℝ.

(I am missing a few for all ε belonging to....and stuff, I just wanted to write it quickly)
Thank you

Last edited:

Related Calculus and Beyond Homework Help News on Phys.org
FactChecker
Gold Member
1) Your definition of differentiation is wrong. It should have h -> 0.
2) You have not used the bound on f''. I think you need that to prove uniform convergence.

1) Your definition of differentiation is wrong. It should have h -> 0.
2) You have not used the bound on f''. I think you need that to prove uniform convergence.
$$f''=\frac{1}{\sqrt{x}}$$