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.

Those binomial coefficients can be visualized in Pascal's triangle as something that looks like a hockey stick, where everything along the shaft representing the left side of your equation, and the blade representing the right side term.

A fundamental property of Pascal's triangle is that the sum of any two numbers in the same row, next to each other, is equal to the number in the row below them, in between them. This is called Pascal's rule.

If you use Pascal's rule $n-r+1$ times, you get the hockey stick identity.

Yes. It can be proved by induction.

In the process of proof, you should know the following equality:

$C_n^r+C_n^{r+1}=C_{n+1}^{r+1}$

Added:

When $n=r$, it holds. Suppose that $n=k$, it also holds. Now let $n=k+1$,

$$C_r^r + C_{r+1}^r + \cdots + C_{k+1}^r+ C_{k+2}^r=C_{k+1}^{r+1}+C_{k+2}^{r}=C_{k+2}^{r+1}$$