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I am trying to prove that a set is closed by induction. Specifically, let me define

Let B_t be sets, and A_T:=sum{B_t: t=1, .., T}=Sum{b_t: b_t in B_t, and t=1, ..., T}

A property that these sets have is that B_s is a subset of B_t for s<=t.

I try to prove A_T is closed by the following argument:

1) First show B_1 is closed.

2) Assume Sum{B_t: t=2, ...., T} is closed.

3) Prove A_T is closed.

My question is whether I can assume that Sum{B_t: t=2, ...T} is closed instead of Sum{B_t: t=1, ...T-1} in 2)

Thank you in advance

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# Proof By Induction

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