Representation of a Function on Natural Numbers Could you help me with an example and explanation on what actually would be a function on $\mathbb{N}$? 
I am told that it can be identified as a sequence $(f_i)$ and it can be written as 
$$f(k)=\sum_{i=1}^\infty \mathbb{1}_{\{i\}}(k)$$
where $\mathbb{1}_i(k)$ is the charachteristic function.
but this seems too abstract and I cannot grasp it. Maybe a simple example will help
 A: I think looking at the example of a finite set might help. Let's say we have a function on the set $\{0,1,2\}$ so that $f(0)=6$, $f(1)=\pi$, and $f(2)=\sqrt{2}$ (these numbers don't actually matter much). We can write $f$ as
$$f(x)=\begin{cases}6&\mathrm{if}\ x=0\\\pi&\mathrm{if}\ x=1\\\sqrt{2}&\mathrm{if}\ x=2.\end{cases}$$
The magic happens because we can write these cases as
\begin{multline*}
f(x)=6\cdot\left[1\mathrm{\ if\ }x=0\mathrm{\ and\ }0\mathrm{\ otherwise}\right]\\
+\pi\cdot\left[1\mathrm{\ if\ }x=1\mathrm{\ and\ }0\mathrm{\ otherwise}\right]\\
+\sqrt{2}\cdot\left[1\mathrm{\ if\ }x=2\mathrm{\ and\ }0\mathrm{\ otherwise}\right],
\end{multline*}
which can be written very concisely as
$$f(x)=6\cdot \textbf{1}_0(x)+\pi\cdot \textbf{1}_1(x)+\sqrt{2}\cdot \textbf{1}_2(x)$$
with indicator functions. What you've written here is no different from this, except that $\mathbb{N}$ is an infinite set instead of a finite set. In general, we can write any function $f:S\to T$ as
$$f(x)=\sum_{i\in S} f(i)\textbf{1}_i(x),$$
as the only nonzero term inside the summation is the one where $x=i$. 
