Simplify and Prove the identity $\sum_{j=k}^n \binom jk = \binom{n + 1}{k + 1}$

Question

I am trying to prove it by algebra.Reference

What i tried so far is to simplify the binomials and cancelling similar variables. I stuck at some point without any clue on this step :

\begin{align*} \sum_{j=k}^n \binom{j}{k} & = \sum_{j=k}^{n} \frac{j!}{k!(j-k)!} & \text{[using the formula for binomial(j, k)]}\\ & = \sum_{j=k}^{n} \frac{j(j-1) \cdot \ldots \cdot (k+1)k!}{k!(j-k)!} & \text{[writing out the factorial]} \end{align*}

I would like for an help to understand if there is any simple way to prove this identity.