Please help me prove by induction the theorm below

Question

let⟨a_n⟩^∞_n=1 and ⟨bn⟩^∞ _n=1 by sequences of numbers

Then ∀ m∈N ∑(a_n+b_n) = (∑a_n) +(∑b_n)

 

Proof by induction

 

Side note: The summations go from n=1 to m

Answer 1

Base step:

 

For m=1, ∑_n=1...1(a_n + b_n) = a₁ + b₁ = ∑_n=1...1a_n +∑_n=1...1b_n

 

Induction step:

 

Assume, for some k ≥ 1, that the property holds when m = k.

 

Then for m=k+1

 

∑_n=1...k+1(a_n + b_n)

 

= ∑_n=1...k(a_n + b_n) + a_k+1 + b_k+1

 

= ∑_n=1...k a_n + ∑_n=1...k b_n + a_k+1 + b_k+1

 

= ∑_n=1...k a_n + a_k+1 + ∑_n=1...k b_n + b_k+1

 

= ∑_n=1...k+1 a_n + ∑_n=1...k+1 b_n

 

= ∑_n=1...m a_n + ∑_n=1...m b_n

 

Q.E.D.