Determine whether the following statements are true of false. If you believe a statement is true, you do not need to justify it. If you believe a statement is false, provide a counterexample. (a) If f : (0,1] → R is continuous, then it is bounded on (0,1). (b) Let f.g: R R be uniformly continuous on R. Then fg is uniformly continuous on R.

(c) If the series – is convergent, then the series -1 an is convergent. (d) Every sequence has a monotone subsequence. (e) Let S be a nonempty subset of R. If sup S E R, then there exists a sequence {en} CS such that en sup S. (f) If {n} is a Cauchy sequence, then the sequence {(-1)”sn} is also Cauchy. (g) Let {en} be a sequence of real numbers. If sn > 0 for every n e N and sn +0, then the series (-1)”sn is convergent.

## Answer

f(0, 1 IR continuous fimution, then it is bounded on (0,1]. This statement is False. Counter example: Define ficool] IR by fat a E01] fiscotinuous on (o. 17 but fo is not bounded. As t Elo7 f (d = n D. Then as at fogair R be uniformly cot on R then fg is uniformly coth on R e This statement is false. Counter example: fag: 1R IR by Suppose Then f g both uniformly oth on R but fog = x n = x is not uniformly coth OnIR © If the series and convergent, then the series of an is also convergent. This statement is false. Counter example: Take, antonen Then an = thor is convergent But it is not convergent. @ Every sean has monotone subsequence This statement is true. using peak we can get this.

e et sa noty subset of R. SupSER clan’s S S such that net sbe a non empty subset ofth then there exists a sequencexans ss such to Bu sups. This statement is true. If supses, then take constant segm An= sups. If sups &s, then using supremum” property I we construct or sean Lores such that on sups. (f) If tnt Camely sequ, then. En ons is also cauchy. This satement is false, Counter example: Choose a constant sega: c) On=lonen. Obviously, fon’s is caudry seam. ut (1) “On = 21, n-even t, n= odd is not a cauchy sean.’ (9) Ket 28n be a segn of real numbers. If so7O, for every nEN and ano, then the series E en is convergente 1. This statement is true by Leibnitz’s test.