Rising Sum of Binomial Coefficients/Proof by Induction

Theorem
Let $n \in \Z$ be an integer such that $n \ge 0$.

Then:
 * $\displaystyle \sum_{j \mathop = 0}^m \binom {n + j} n = \binom {n + m + 1} {n + 1} = \binom {n + m + 1} m$

where $\displaystyle \binom n k$ denotes a binomial coefficient.

That is:
 * $\displaystyle \binom n n + \binom {n + 1} n + \binom {n + 2} n + \cdots + \binom {n + m} n = \binom {n + m + 1} {n + 1} = \binom {n + m + 1} m$

Proof
Proof by induction:

Let $n \in \Z$.

For all $m \in \N$, let $P \left({m}\right)$ be the proposition:
 * $\displaystyle \sum_{j \mathop = 0}^m \binom {n + j} n = \binom {n + m + 1} {n + 1}$

$P(0)$ is true, as this just says:
 * $\displaystyle \binom n n = \binom {n + 1} {n + 1}$

But $\displaystyle \binom n n = \binom {n + 1} {n + 1} = 1$ from the definition of a binomial coefficient.

Basis for the Induction
$P(1)$ is the case:

So:
 * $\displaystyle \sum_{j \mathop = 0}^1 \binom {n + j} n = \binom {n + 2} {n + 1}$ and $P(1)$ is seen to hold.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 1$, then it logically follows that $P \left({k+1}\right)$ is true.

So this is our induction hypothesis:
 * $\displaystyle \sum_{j \mathop = 0}^k \binom {n + j} n = \binom {n + k + 1} {n + 1}$

Then we need to show:
 * $\displaystyle \sum_{j \mathop = 0}^{k+1} \binom {n + j} n = \binom {n + k + 2} {n + 1}$

Induction Step
This is our induction step:

So $P \left({k}\right) \implies P \left({k + 1}\right)$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\displaystyle \sum_{j \mathop = 0}^m \binom {n + j} n = \binom {n + m + 1} {n + 1}$

Finally, from Symmetry Rule for Binomial Coefficients:
 * $\displaystyle \binom {n + m + 1} {n + 1} = \binom {n + m + 1} m$