國外友人請我幫問數學題目- 6

Question

6. Let S(n) and T(n) be bounded sequences. Prove that lim inf(S(n) + T(n))≧ lim inf S(n) + lim inf T(n). Produce an example in which the inequality is strict.

Answer 1

最佳解答

Let

An = { T(i) + S(i) : i is integer and i ≧ n }

Bn = { T(i) : i is integer and i ≧ n }

Cn = { S(i) : i is integer and i ≧ n }

1. claim: inf An ≧ inf Bn + inf Cn.

For each i ≧ n, T(i)≧inf Bn and S(i)≧ inf Cn.

Then S(i)+T(i) ≧ inf Bn + inf Cn for each i ≧ n.

It means inf Bn + inf Cn is a lower bound of An.

So we have inf An ≧ inf Bn + inf Cn.

Taking limit on both sides, it means lim inf An ≧ lim inf Bn + lim inf Cn.

2. Let S(n)=(-1)^n, T(n)=(-1)^(n+1). We have S(n)+T(n)=0 for all n.

Moreover, lim sup S(n) = lim sup T(n) = - 1

Then lim sup S(n)+T(n) = 0 > - 2 = lim sup S(n) + lim sup T(n)