1. (a) Show that lim n →∞ k n /n! = 0 for all k ∈ . (b) What can be said about limn →  n!/k n ? 2. Suppose that (s n ) is a convergent sequence with a ≤ s n ≤ b for all n ∈ . Prove that a ≤ lim s n ≤...

1. (a) Show that lim_{n →∞}
kⁿ/n! = 0 for all k ∈
.

(b) What can be said about limn → n!/kⁿ
?

2. Suppose that (s_n) is a convergent sequence with a ≤ s_n
≤ b for all n ∈
. Prove that a ≤ lim s_n
≤ b.

3. Prove the following.

(a) If lim s_n
= +
 and k > 0, then lim ks_n
= +
.

(b) If lim s_n
= +
 and k <>_n
= −
.

(c) lim s_n
= +
 iff lim (− s_n) = −
.

(d) If lim s_n
= +
 and if (tn) is a bounded sequence, then lim (s_n
+ t_n) = +
.

(e) If (s_n) converges to L > 0 and lim t_n
= +
, then lim (s_nt_n) = +
.