It is known that in the positive term sequence {an}, for all n ∈ n *, there is an & # 178; ≤ an-a (n + 1 holds) 1. Prove: sequence It is known that in positive term sequence {an}, for all n ∈ n *, there is an & # 178; ≤ an-a (n + 1 holds). ① prove that any term in sequence {an} is less than 1. ② explore the size of {an} and 1 / N, and prove your conclusion. (mathematical induction)

It is known that in the positive term sequence {an}, for all n ∈ n *, there is an & # 178; ≤ an-a (n + 1 holds) 1. Prove: sequence It is known that in positive term sequence {an}, for all n ∈ n *, there is an & # 178; ≤ an-a (n + 1 holds). ① prove that any term in sequence {an} is less than 1. ② explore the size of {an} and 1 / N, and prove your conclusion. (mathematical induction)

∵ (an) & #178; ≤ an-a (n + 1), a (n + 1) ≤ an - (an) & #178;
∵ in the sequence {an}, an ＞ 0,
∴a(n+1)＞0,
∴an-(an)²＞0,
∴0＜an＜1
So any term in the sequence {an} is less than 1
（2）
From (1), we know that 0 < an < 1 and a (n + 1) ≤ an - (an) & # 178;
Then A2 ≤ A1 &; (A1) &; = &; (A1 &; 1 / 2) &; + 1 / 4 ≤ 1 / 4