Question: 1) An infinite decimal x = do.ajd2 ... is eventually periodic if there are positive integers n and k such that ditt = a; for

1) An infinite decimal x = do.ajd2 ... is eventually periodic if there are positive integers n and k such that ditt = a; for all i > n. Show that any decimal expansion which is eventually periodic represents a rational number. HINT: Compute 107+x - 10"x. 2) Show that the sequence (logn),=] does not converge. 3) (a) Show that ! 0 for all 5) " > 1. Prove that the sequence either converges or tends to +co. For the following sets, find the supremum and infimum. Which have a max or min? (@) A = {ata-': aEQ, a>0). (b) B = {a+ (20)-1: aE Q, 0.1

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