I was reviewing linearity of expectations in my textbook and saw the following statement (see arrow in picture).
I understand how the theorem can be proved for multiple random variables using induction, but was wondering: is it necessary to use induction? It seems to me that it's fairly easy to generalize the proof for two random variables for multiple random variables without needing to go through the steps of induction. Asking because I saw another source also saying that linearity of expectation for multiple random variables can be proved by induction. Just wanted to make sure I'm not missing any subleties here. Screenshot here
Best Answer
The thing is you are interested in demonstrating the proposed property for every natural number.
If you restrict your proof to some $n\in\mathbb{N}$, you cannot conclude it holds for every $n\in\mathbb{N}$.
That is why the proof by induction is important.