4. For n 0; 1; 2; 3; . . . , consider the sequence defined by ym 1 n1! ; m 3n,

4. For n ¼ 0; 1; 2; 3; . . . , consider the sequence defined by ym ¼

1

ðnþ1Þ! ; m ¼ 3n,

ð1Þn10 þ ð1Þnþ1n 2ðnþ1Þ

; m ¼ 3n þ 1,

ð1Þnþ1 þ ð1Þn 10ðnþ1Þ

; m ¼ 3n þ 2.

(a) Determine all the limit points of this sequence and the associated convergent subsequences.

(b) Determine the formula for Un and Ln, as given in the definition of limits superior and inferior, and evaluate the limits of these monotonic sequences to derive lim sup ym and lim inf ym, respectively.

(c) Confirm that the limit superior and limit inferior, derived in part (b), correspond to the l.u.b. and g.l.b. of the limit points in part (a).