By Nikolsky S.M.

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Not bounded below the f¡0 11 • t um e~ we denote 1t b) but Is • • owmg wo cases are posstble: Case_ l. There Is a fimte number a< b such that the ineq l't' b are sattsfied fo · fi 't ua 1 tes a ~ Xn ~ . sub ese va uefs natural numbers {n } - {n < } . sequence o th b { k 1 n2 < .... s~quenc~ -~nJ _of. the ori~inal sequence which is obvious]y bounded proev~~~\~~~~c~dflyes~:~~~~f~nor of such a subsequence has aiready bee~ C'Q' n2 so that x 83 If a sequence is not bou~ded abov~ it is clear that it contains a subsequence convergent to + and smce + IS greater than any number we ha ve n3 > LlMIT OF SBQUENCE finite number of elements Xn.

Not c~n­ taining x 1• Again, according to the dcfinition of lhe pomt a. thcre IS a pomt bclonoing to the latter interval such that x2 E E and X2 ~ a. Next we can find a~ i~terval (e 2 , c/2) of length d2-c2 < 1/2, containing the point a and oot co ntaining x 1 and x 2, in which thcre is a point xa E E such that xa ;t: a, etc. As a result, we obtain t he required sequcnce. Thus, thc dcfinition of a limit point can be restated in the cquivalent alternative form: a point a is a limit point of a set E if its any neighbourhood contains an infillitude of pvints belonging toE.

Hence*"*, li =(J+e )" ~~- - 2. At the samc time, WC ha ve lim 0 H, 11' . k __ et:ausc thc mequalities ¡tn ::-. N and n > Nk (wherc N> O) imply each other, ami thcreforc oivcn an}· · ( k ' ~ • · ts no namely, any IICJ > Nk) such that •~"ñ N r ll r > 1 or a n > n . ~ l. li m Let a ~ O ~nd let k he a natural number. - ~a the contrary JS Statcd-, the arithmetic kth root of a~~· ~h~ll. me~nl. ss num_ber whose kth power is equalto a. Such a numb~r a . ml! It wlll be more :onveniem lo provc rhis assertion lat e(~~tt :nd lS un_Jqu~.

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Course in Mathematical Analysis by Nikolsky S.M.

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