Real Analysis and Applications by Fabio Silva Botelho

Author:Fabio Silva Botelho
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

and

Therefore,

so that

Hence,

Since ε > 0 is arbitrary, we may infer that

The proof is complete.

Also very important is the next result.

Proposition 7.4.4

Let . Under such hypotheses, we have

Proof

If m ∗(A) = +∞ or m ∗(B) = +∞, the result follows immediately, since in such a case,

Thus, suppose m ∗(A) = α 1 < +∞ and m ∗(B) = α 2 < +∞.

Let ε > 0.

Since α 1 + ε∕2 > α 1 and α 2 + ε∕2 > α 2 there exist sequences {I m } and {J m } of open real intervals such that

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