## Uniform convergence

For college and university level advanced Mathematics
nayel
Posts: 268
Joined: Tue Dec 07, 2010 7:38 pm
Location: Dhaka, Bangladesh or Cambridge, UK

### Uniform convergence

Determine whether the sequence of functions $\displaystyle f_n(x)=e^{-x^2}\sin\frac xn$ converges uniformly on $\mathbb R$.
"Everything should be made as simple as possible, but not simpler." - Albert Einstein

tanvirab
Posts: 446
Joined: Tue Dec 07, 2010 2:08 am

### Re: Uniform convergence

for any $\epsilon$ we can choose $x_0$ = $inf\{x : e^{-x^2} < \epsilon \}$
now we can choose $n$ large enough so that $sin(\frac{x}{n}) < \epsilon$ for all $|x| \leq x_0$

nayel
Posts: 268
Joined: Tue Dec 07, 2010 7:38 pm
Location: Dhaka, Bangladesh or Cambridge, UK

### Re: Uniform convergence

Does it not need to hold for all $x\in\mathbb R$?
"Everything should be made as simple as possible, but not simpler." - Albert Einstein

tanvirab
Posts: 446
Joined: Tue Dec 07, 2010 2:08 am

### Re: Uniform convergence

It does (right?)

For $|x| \leq x_0$ the $sin$ part is smaller than $\epsilon$ and for $|x|>x_0$ the $exp$ part is smaller than $\epsilon$. So the product is always smaller than $\epsilon$. (?)

nayel
Posts: 268
Joined: Tue Dec 07, 2010 7:38 pm
Location: Dhaka, Bangladesh or Cambridge, UK

### Re: Uniform convergence

That looks so easy compared to my one page long solution!
"Everything should be made as simple as possible, but not simpler." - Albert Einstein

tanvirab
Posts: 446
Joined: Tue Dec 07, 2010 2:08 am