Proof of Closed Set by Induction

[submartingale]

Hello all,

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

 

[mmwave]

for any help or suggestions!

Hello,

Thank you for sharing your approach to proving that A_T is closed by induction. Your argument seems sound, but I would suggest being more specific in your notation. For example, instead of using "Sum" notation, you could use the sigma symbol (Σ) to denote summation. Also, in your third step, it would be helpful to clarify how you plan to use the assumption from the second step in your proof.

To answer your question, it is generally acceptable to use the assumption that Sum{B_t: t=2, ..., T} is closed instead of Sum{B_t: t=1, ..., T-1} in your proof. This is because, if the set is closed for all t=2, ..., T, then it must also be closed for t=1. However, it would be a good idea to include a brief explanation of why this assumption is valid in your proof.

I hope this helps. Good luck with your proof!