Solution of the 5th Homework
نویسنده
چکیده
Let un sup(ak )∞ k n and vn sup(bk ) ∞ k n be suprema. Then the following obvious relation ak ≤ un and bk ≤ vn , ∀k ≥ n ⇒ ak + bk ≤ un + vn , ∀k ≥ n shows that, upon taking the supremum over all k with k ≥ n, we get sup(ak + bk )∞ k n ≤ un + vn . This shows that for any n ≥ m, we get lim sup k→∞ (ak + bk ) inf j: j≥m sup(ak + bk )∞ k j ≤ sup(ak + bk ) ∞ k n ≤ un + vn Since we know that (un )∞ n m and (vn )∞ n m converge to lim supn→∞ an and lim supn→∞ bn respectively, taking n →∞ gives lim sup n→∞ (an + bn ) ≤ lim sup n→∞ an + lim sup n→∞ bn as desired.
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