Problem: Graph the following functions and formally justifying your answer. \begin{eqnarray*} \sum_{n=1}^{\infty}\chi_{[n,+\infty)} \quad \text{and} \quad \sum_{n=1}^{\infty}\chi_{[0,n]} \end{eqnarray*}
My approach: By definition, we know that if si $E\subset \mathbb{R}$, eso characteristic function of $E$ let's to define as $$\chi_{E}(x):=\left\{ \begin{aligned} 1, \quad x \in E\\ 0, \quad x\notin E\end{aligned} \right.$$ Then, by definition we can see that \begin{eqnarray*} \chi_{[n,+\infty)}(x)=\left\{ \begin{aligned} 1, \quad x \in [n, +\infty)\\ 0, \quad x \notin [n,+\infty) \end{aligned}\right. \quad \text{y} \quad \chi_{[0,n]}(x)=\left\{ \begin{aligned} 1, \quad x \in [0, n]\\ 0, \quad x \notin [0,n] \end{aligned}\right. \end{eqnarray*} So, \begin{eqnarray*} \sum_{n=1}^{+\infty}\chi_{[n,+\infty)}(x)=\chi_{[1,+\infty)}(x)+\chi_{[2,+\infty)}(x)+\chi_{[3,+\infty)}(x)+\cdots \end{eqnarray*} and \begin{eqnarray*} \sum_{n=1}^{+\infty}\chi_{[0,n]}(x)=\chi_{[0,1]}(x)+\chi_{[0,2]}(x)+\chi_{[0,3]}(x)\cdots \end{eqnarray*}
But, I don't know how to continue. How can I grah that functions? Is there another approach for to solve this problem? Thank you!
This is a plot of the function $$\sum_{n=1}^{+\infty}\chi_{[n,+\infty)}(x)$$