Ha, men Bolzano-Veyerstrass teoremasining oddiy isbotini taqdim eta olaman. Mana u: Bolzano-Veyerstrass teoremasi shuni ko'rsatadiki, real sonlarning har qanday chegarali ketma-ketligi konvergent quyi ketma-ketlikka ega. Dalil: {a_n} real sonlarning chegarali ketma-ketligi bo'lsin. Bu shuni anglatadiki, M > 0 shunday M mavjud. a_n <= M barcha n uchun. Endi biz quyidagicha ikkita ketma-ketlikni qurishimiz mumkin: 1. B_1 = a_1 2. N > 2, b_n = a_k, bu erda k eng kichik indeksdir, shunda a_k {b_1, b_2,..., b_n-1} to'plamda mavjud emas. 3. C_n = {a_1, a_2, a_
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