By A. A. Milyutin and N. P. Osmolovskii

The speculation of a Pontryagin minimal is constructed for difficulties within the calculus of diversifications. the appliance of the inspiration of a Pontryagin minimal to the calculus of adaptations is a particular characteristic of this e-book. a brand new thought of quadratic stipulations for a Pontryagin minimal, which covers damaged extremals, is built, and corresponding enough stipulations for a powerful minimal are bought. a few classical theorems of the calculus of diversifications are generalized

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We can easily see, however, that this difficulty is not serious. It happens only in decimal fractions in which the period consists of 9s only. Otherwise two different decimal fractions cannot have the same "value," and two different "values" cannot belong to one and the same decimal fraction. The results of this discussion incidentally show in an altogether new way that there are irrational numbers. 10100100010000 ... , in which the l's are separated successively by one, two, three, four, ...

For all n we have 1st- sntnI +(Snt- Sntn)I Sn) + sn(t - tn) I It(s - Sn)I + ISn(t - tn) I It I Is- s; I + IsnI It - tn I · I(st It(s ~ ~ Snt) What we have to prove now is that for any preassigned E > 0 an r can be found such that 1st- sntnI ~ E for any n ~ r, Now let E be given. From Theorem 2 we know that the convergent sequence Snis bounded-that there exists a bound M such that, for all n, ISnI ~ M. The same is true for the sequence i; We choose M positive and large enough to be a bound for both sequences.

For if, for each M there were some 11/s < (l/M); that is, for every E = l/M n l > M, then ISnl there would be some Sn such that ISnI < E. 0 . n-'OO Hence there is some M such that, for all n, Il/s" I~ M. Next, 1I Sn- S I 1 1 1 S1 - Sn = I~ = 1sT I s" - S ITS:T ~ 1sTI s" - S 1M. 00 proach zero. Hence we can choose a number p such that, for n I s" - S I<~ M . ~ S I will ap- p, we have 38 THE CALCULUS Then I(lis) - (lis,,) be proved. THEOREM: 5. oo Ii m til = t . > t. Now the following holds: t - s = (t - t,,) + (t" - s,,) + (s" - Proof.