Let en and ty be bounded sequences of nonnegative numbe and stProve lim sup sutn lim sup sn lim sup tn

We can use the definition of limit superior to prove this inequality. Recall that:

• lim sup an = lim n→∞ sup{k≥n} ak, where sup{k≥n} ak denotes the supremum (least upper bound) of the subsequence {ak : k ≥ n}.
• Similarly, lim sup bn = lim n→∞ sup{k≥n} bk.

Let cn = sn tn be the sequence of products of the terms in (sn) and (tn). Then:

• lim sup cn = lim n→∞ sup{k≥n} sn tn = (lim sup sn) (lim sup tn), by the product rule for limit superior.

• Moreover, for any n ≥ 1, we have:

  • sup{k≥n} cn = sup{k≥n} sn tn ≤ sup{k≥n} sn sup{j≥n} tj, by taking the supremum over k and j separately.
  • Therefore, lim n→∞ sup{k≥n} cn ≤ (lim sup sn) (lim sup tj), by taking the limit as n → ∞.

Combining these two results, we obtain:

lim sup cn < (lim sup sn) (lim sup tn),

as desired. Note that the strict inequality holds because the sequences (sn) and (tn) are assumed to be bounded and nonnegative, so their limit superior is finite