Define a function such that lim f n = f pointwise (Homework help)

Question

I'm not sure how to define this function, do I just say f(a)=0, then apply the definition to show it converges pointwise. When I graph increasing values of n it seems like the function is becoming the y-axis.

So my question is does f(a)=0 define this piecewise function as n approaches infinity?

Any help would be greatly appreciated. Thanks,

Answer 1

Let $a\in [0,1] $

If $a=0$ then for $n>0$ we have

$f_n (0)=1$ thus $\lim_{n\to+\infty}f_n (0)=1=f (0)$

If $0 <a\le 1$ then for $n $ large enough

$$\frac {1}{n}<a\le 1$$ and $f_n (a)=0$.

$$\lim_{n\to+\infty}f_n(a)=0=f (a)$$

The function pointwise limit is defined for $x\in [0,1] $ by $f (x)=0$ if $0 <x\le 1$ and $f (0)=1$