(related to Proposition: The General Perturbation Method)
Since the sets \(\{1,2,\ldots,n\}\) and \(\{n+1\}\), respectively \(\{0\}\) and \(\{1,2,\ldots,n+1\}\) of natural numbers and disjoint, by applying the rule of combining different sets of indices we get two (trivial) splits of the sum \(S_n\):
\[S_n + a_{n+1}=\sum_{0\le k\le n+1} a_k= a_0 + \sum_{1\le k\le n+1} a_k.\]
By shifting the index of the right side of the equation twice ("perturbation" of the indices, which gives this method its name) we further get
\[a_0 + \sum_{1\le k\le n+1} a_k=a_0 + \sum_{1\le k+1\le n+1} a_{k+1}=a_0 + \sum_{0\le k\le n} a_{k+1},\]
which completes the proof.
