How to prove this statement: $\binom{r}{r}+\binom{r+1}{r}+\cdots+\binom{n}{r}=\binom{n+1}{r+1}$ Let $n$ and $r$ be positive integers with $n \ge r$. Prove that

Still a beginner here. Need to learn formatting.
I am guessing by induction? Not sure what or how to go forward with this.
Need help with the proof.

What is the number on the left counting?
The number of subsets of size $r$ of $\{1,2,3\dots r\}$ plus the number of subsets of size $r$ of $\{1,2,3\dots r+1\}$ plus the number of subsets of size $r$ of $\{1,2,3\dots r+1,r+2\}$ and so on up until the number of subsets of size $r$ of the set $\{1,2,3\dots n\}$
What is the element in the right counting? Th number of sunsets of size $r+1$ of the set $\{1,2,3\dots n+1\}$
Lets give a bijection between the elements in the right and those in the left.
How? suppose you have a subset $S$ of size $r+1$ of $\{1,2,3\dots n+1\}$ Let $k$ be it's greatest element, we associate to this subset a subset of size $r$ of the set $\{1,2,3\dots k-1\}$ which one?
$S-k$ Clearly this is a function from the obects counted on the right to those counted on the left. It is injective since different subsets go to different sets, and it is surjective since the subset $R$ of size $r$ of $\{1,2,3\dots m\}$ comes precisely from the set $R\cup \{m+1\}$.
