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## 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:

f

_{n}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 f

_{n}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

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