An inductive definition needs a base case (B_1).

For the proof, it's good to say that induction is on n (not on t).

P was not defined above.[Base Case]:

Prove P(1)

Let n=1

Fix (consider) t∈ℕ s.t 0≤t<2^1{2^0}⊆B_1 ∑[x∈{2^0}]x = 1What is W.T.P.: Want To Prove? Also, write "Case 1" instead of "C1."[I.S]:

Let n≥2

Suppose P(n) [I.H]

W.T.P P(n+1) i.e. t∈ℕ s.t 0≤t<2^(n+1) → ∃ subset A s.t A⊆B_(n+1) ˄ ∑[x∈A]x = t

This is the induction hypothesis.t∈ℕ s.t 0≤t≤2^(n)-1 → ∃ subset A s.t A⊆B_n ˄ ∑[x∈A]x = t

After that, it became too hard to understand. Are you claiming these facts or are you going to prove them? What does "for n" mean? Rearranging which inequality? A formal proof does not have to be difficult to read.