Question

For each of the following statements, either prove it is true, or provide a counterexample to show that it is false. (a) If (sn) is a sequence such that lim sn = 0, then lim inf|sn= 0. (b) If f : [0, 1] + R is a function with f(0) < 0 and f(1) > 0, then there exists CE (0,1) such that f(c) = 0. (c) If I is an interval, f:I + R is continuous on I, and (2n) is a Cauchy sequence in I, then (f(n)) is a Cauchy sequence in R.

Answer 1

First we prove that if Lim sn=0 then 4 Lim, snl zo - Proof Let 67o. Since Limsno there exista natural number k such that is niet & nyk. I As lisnl-ol = lsnl it follows that it Iisal-01 <t # nyk This proves that Lith Isnl = 0 Now we know that if a sequence {nn) is convergent then Liman = Liman = Lim an. nita Hence, since Lim Isalto. Thus Lim inf 1sn1 = 0. Thus Lim infisala. T This statement is L true].

b) consider the function P: [0, 1] lk defined by fix) = -1 when osat = 1 when I sa cl. we see that fco) = -1 and f(1) = 1. ie fro) = -1 <0 and f(1) = 1 70 then those existed of open but there does not exist any ef(0,1) such that f(e)=o. Hence the statement is false. > Consider the function f: (0,1) R defined by fra) = sint 16,0,1). Let an= 2 & nein which is a cauetry sequence in Pono (0,1). but the sequence {f } is 4,9-1,0,--...] which is a divergent sequence and this is not a caucty sequence. Thus the given statement is false