12. For n 0; 1; 2; 3; . . . , consider the sequence defined by...

12. For n ¼ 0; 1; 2; 3; . . . , consider the sequence defined by xm ¼

ð1Þn nþ1 ; m ¼ 5n, 1 þ ð1Þnn 2ðnþ1Þ

; m ¼ 5n þ 1,

1 þ ð1Þn nþ1 ; m ¼ 5n þ 2,

n2 þ n; m ¼ 5n þ 3, 10en; m ¼ 5n þ 4.

8>>>>>>>><

>>>>>>>>:

(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 xm and lim inf xm, 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).