# Cauchy sequences of compactly supported functions

We have the three standard norms $$\left\Vert\right\Vert_1$$, $$\left\Vert\right\Vert_2$$, $$\left\Vert\right\Vert_\infty$$ defined on the space of compactly supported functions $$C_0(\Bbb N,\Bbb R)$$, $$f: \Bbb N \rightarrow \Bbb R$$.

For each pair of norms, I must to find sequences which are cauchy with respect to one of these norms but not the other. For instance, in considering the $$\left\Vert\right\Vert_1$$ and $$\left\Vert\right\Vert_2$$ norms, I must find a sequence of functions that is cauchy for $$\left\Vert\right\Vert_1$$ but not $$\left\Vert\right\Vert_2$$, for the $$\left\Vert\right\Vert_1$$ and $$\left\Vert\right\Vert_\infty$$ norms I must to find a sequence of functions that is cauchy for $$\left\Vert\right\Vert_1$$ but not $$\left\Vert\right\Vert_\infty$$, and for the $$\left\Vert\right\Vert_2$$ and $$\left\Vert\right\Vert_\infty$$ norms I must find a sequence of functions that is cauchy for $$\left\Vert\right\Vert_2$$ but not $$\left\Vert\right\Vert_\infty$$ or vice versa.

I have attempted many approaches but I am having difficulty building an intuition for the behavior of these norms in the context of sequences of functions. Any guidance would be much appreciated.

Note that $$\Vert \cdot \Vert_\infty \leqslant \Vert \cdot \Vert _1$$ and $$\Vert \cdot \Vert_\infty \leqslant \sqrt{\Vert \cdot \Vert _2}$$ and when $$\Vert f \Vert_1 <1$$, then $$\Vert f \Vert_2 \leqslant \Vert f \Vert_1$$. So any Cauchy sequence for $$\Vert \cdot \Vert _1$$ or for $$\Vert \cdot \Vert _2$$ will be a Cauchy sequence for $$\Vert \cdot \Vert _\infty$$, and any Cauchy sequence for $$\Vert \cdot \Vert _1$$ will be a Cauchy sequence for $$\Vert \cdot \Vert _2$$
For the other cases, basically use the sequences of sequences $$(f_N)_{N\in \mathbb{N}} \text{ with } f_N=(1,1/2,\cdots,1/N,0,0,0,\cdots)$$ and $$(g_N)_{N\in \mathbb{N}} \text{ with } g_N=(1,1/\sqrt{2},\cdots,1/\sqrt{N},0,0,0,\cdots).$$
Hint: Let $$E_k=\{1,\dots ,k\}$$. Then let $$f_{k}(n):=\frac{1}{n}1_E(n)$$. So for example, $$f_1(1)=1$$, $$f_1(n)=0$$ for $$n>1$$. Then $$f_2(1)=1$$, $$f_2(2)=\frac{1}{2}$$, $$f_2(n)=0$$ for $$n>2$$. Then $$f_3(1)=1$$, $$f_3(2)=\frac{1}{2}$$, $$f_3(3)=\frac{1}{3}$$, $$f_3(n)=0$$ for $$n>3$$. And so on. Now let $$f(n):=\frac{1}{n}$$. What types of convergences are happening?