Finding the pointwise limit and checking for uniform convergence

Question

I am wondering if I'm thinking properly regarding this question:

First, according to the pointwise limit, I did this informally:

As $n$ becomes very large, the interval $\left[\frac {1}{n}, \frac {2}{n}\right]$ will contain no $x$.

Clearly, $f_n(0)=0$.

My question here is can we consider the last interval as $x>0$? If this is right so we have $f_n$ converges pointwise to the constant zero function.

Regarding uniform convergence, we can find max $\vert f_n(x) - f(x) \vert $ on the first interval getting $1$ and not zero so this doesn't converge uniformly. Right?

Anyone can help me?

And finally, can you help me finding more exercises like this ? I have an exam and wanna be stronger in dealing with such exercises. Thanks in advance.

Answer 1

You have the right ideas. To be a bit more formal: Clearly $f_n(0) \to 0$. If $x > 0$, then by the Archimedean property of the real numbers, we can find a natural number $n$ such that $2/n < x$. As $f_m(x) = 0$ for all $m \geq n$, we conclude that $f_n(x) \to 0$, and hence that $f_n$ converges pointwise to the zero function. However $\sup_x |f_n(x)| = 1$ for all $n$, and so convergence is not uniform.

Answer 2

For all $n>0$

$$f_n (\frac {1}{n})=1$$ thus

$$\sup_{\mathbb R^+}|f_n (x)-0|\geq 1$$

the sequence doesn't converge uniformly to zero function.