I have just finished highschool and have started learning on my own some combinatorics and how to do proofs, and while messing around with sums and Pascal's triangle I found an interesting yet trivial property that I tried to prove. I'm assuming that it has already been found, but I couldn't find anything mentioning it.
So my questions are: Is my proof correct? Has this property been found already? What can this property be used for?
Let $$f(x,n)= \sum_{i_1=1}^n \sum_{i_2=1}^{i_1} \sum_{i_3=1}^{i_2} \cdots \sum_{i_x=1}^{i_{x-1}}i_x \ $$ where $x$ is the number of sigma sums.
My conjecture is that $$f(x,n)={n+x \choose x+1}$$
Proof by induction
Basis step:
$$\begin{align}
f(1,n) &=\sum_{i=1}^ni\\
&=\frac{n(n+1)}{2}\\
&={n+1 \choose 2}
\end{align}$$
The basis step works. This first result is already proven for all n.
Inductive step: Assume that $f(x,n)= {n+x \choose x+1} $
Now, $$\begin{align}f(x+1,n) &=\sum_{m=1}^n \sum_{i_1=1}^{m} \sum_{i_2=1}^{i_1} \cdots \sum_{i_x=1}^{i_{x-1}}i_x\\
&=\sum_{m=1}^n f(x,m)\\
&=\sum_{m=1}^n {m+x \choose x+1}\\
&=\sum_{l=0}^{n-1}{l+1+x \choose x+1}\\
&={n+x+1 \choose x+2}\\ \end{align}$$
Therefore, if $f(x,n)$ holds, then $f(x+1,n)$ holds.
What I am not sure about is whether I have to use induction to prove that this property holds for every integer $n$ as well,