By Nikolsky S.M.

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Extra info for Course in Mathematical Analysis

Sample text

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.

S+ belfs. -: ~ut th~ hm Xn lim Xn =- r: and we have thus proyed t~at in Case 1 lim Xn also exists. Case 2. ualilty a'~ Xn holds :nfi~ite ~~m~~;c~/~:~:e:a~;e~~ · ts o vtous y means that n-eo lim If a sequence is bounded above by a finite n b ( . 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 < } .

2! __ 3! + ... 1+ . • . +-1-<2+1=3 zn-l " = .... _) (t-2)+ ... ' 2! 3! (1) < ; (n >N) 11 < e -f and lxm-al < e 2 e 2 e +2 = f *A. L. ) in tbe way characteristic of modern mathematics. x,-a¡ Let n and m be any two natural numbers exceeding N. Then lx,-aj Let a variable x, (n = 1, 2, ... 6. 10 we shall show how the number e can be evaluated with an arbitrary preassigned accuracy. . After the notion of a limit of a funct10n has been mtroduced we s~all ~ee that the above limit exists and is equal to e even when n tends to plus mfimty and to minus infinity assuming continuously the corresponding real values.