- #1
AllRelative
- 42
- 2
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: