Question

Separate each answer?

5) Define the supremum of a bounded above set SCR. 6) Define the infimum of a bounded below set SCR. 7) Give the completeness property of R 8) Give the Archimedean property of R. 9) Define a density set of R. 10) Define the convergence of a sequence of R and its limit. 11) State the Squeeze theorem for the convergent sequence. 12) Give the definition of increasing sequence, decreasing sequence, monotone se- quence. 13) Give the definition of subsequence 14) State the Bolzano-Weirstrass theorem. Exercise 2 Show that does not exist a rational number r such that 2 = 3 Exercise 3 Let S = {1-1/n, ne N} U{-1+1/n, n e N} . Show that S is bounded and find, sup(S) and inf(S). Exercise 4 Use the definition to prove that lim2n = 2 n+12 Exercise 5 Use the squeeze theorem to determinate limon where sin(na + 2n +1) n+1 Exercise 6 Compute limul and limon for the sequence (ann given by In =

Answer 1

A Supremum ? Let sbe a non emply Subset of R , its is bounded above and - S has a least upped bound then call -----citas supremým of S. Infimum' let sbe a non emply Subset of R it s es bounded below and S has greatest lowed bound then call itas ofimum of se - 6 Completeness Property of R Every nonemple subiet is of R that is bounded above has least upped bound. 6) Archimedean Popedly :- IF ato and bro then for some positive integed a we have nazb. 3 density set of R: Let ☆ be subset of R, and y subuel ofx. Then Y is said to be dense Inx" ir for every xEX there is sequence. Yg) of numbers inr such that yo) convedges bon o conveegenee of sequcence of R. I limit - A sequence ging of real numbers is said to converge to real numbed x provided For each Ero £ N such that - non an-x/CE. If (xo converges for then n is called limit of sequence (an).

of b rn. La Squeeze theorem: If can, Co2 be hoo sequences Convedging to l and Cond is sequence with an abne ca fnad, nando then (bn) also convedges todo o increasing sequence: A sequence (an) of real numbers is increasing it un sant1 fn. . and (en) es decreasing it HADHAH A sequence which es increasing or decreasing will called as 19 moponie sequence 9 subsequence : A subsequence of sequence Cendre és sequence of form (tlked where fk then is positive integer nk such that n. cmc.cnk ch kyc and tr=unk. Bolzano- Heirs tra's Theeremia fuery bounded sequence has convergent -- Subsequence