Prove that $\sum_{i = m}^n a_i + \sum_{i = n + 1}^p a_i = \sum_{i = m}^p a_i$, where $m ≤ n<p$ are integers, and $a_i$ is a real number assigned to each integer $m ≤ i ≤ p$. (Hint: you might want to use induction)
Let's follow the hint and use induction on $p-m = k$
Base case: $k = 1$, then $p = m + 1$ and $n = m$. $\sum_{i = m}^n a_i + \sum_{i = n + 1}^p a_i = a_m + a_{m+1}$ and $\sum_{i = m}^p a_i = a_m+a_{m+1}$. Therefore, the right-hand side is equal to the left-hand side.
Inductive step: Assume for $p-m=k$ the statement holds, show for $p-m = k + 1$. We know that $\sum_{i = m}^n a_i + \sum_{i = n + 1}^{m+k} a_i = \sum_{i = m}^{m+k} a_i$. Now, $\sum_{i = m}^n a_i + \sum_{i = n + 1}^{m+k+1} a_i = \sum_{i = m}^n a_i + \sum_{i = n + 1}^{m+k} a_i + a_{m+k+1} = \sum_{i = m}^{m+k} a_i + a_{m+k+1}$ by inductive hypothesis. Therefore, we get $$\sum_{i = m}^n a_i + \sum_{i = n + 1}^{m+k+1} a_i =\sum_{i = m}^{m+k+1} a_i$$
Is this prove plausible. At this point about the finite sum, I can use the following facts:
if $ m < n \sum_n^m a_i= 0$
if $n \ge m - 1 \sum_{m}^{n+1}a_i = \sum_{m}^{n}a_i + a_{n+1}$