MHB Prove this fibonacci sequence w/o binet formula

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The discussion revolves around proving the Fibonacci identity \(u_{m+n}=u_{m-1}u_n+u_mu_{n+1}\) for natural numbers \(m\) and \(n\) without using the Binet formula. The original poster expresses confusion about where to start and seeks guidance on how to approach the proof without relying on the Binet formula. Suggestions include using mathematical induction on \(n\) as a potential method for the proof. There is also a query about whether the derivation of the Binet formula should be included in the proof process. The focus remains on finding a valid proof strategy for the Fibonacci sequence identity.
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I have a problem and honestly have no idea even where to start. I've been staring at it and thinking about it for over 24 hours...
Let \(u_n\) denote the \(n^{th}\) Fibonacci number. Without using the Binet formula for \(u_n\), prove the following for all natural numbers \(m\) and \(n\) with \(m\geq{2}\):
$$u_{m+n}=u_{m-1}u_n+u_mu_{n+1}$$

I have gone through a couple proofs regarding the Fibonacci numbers in this class before, but never one with two unknowns, and also never one where we weren't able to use the Binet formula. Without the Binet formula, what do I have to work off of? Do I need to write the derivation for the Binet formula within this whole proof?
 
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Try induction on $n$.
 

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