Problem. Given a sequence {tn}, suppose
lim n→∞
tn = T, where T > 0.
Prove that
1
tn =
T .
(Hint: You want to estimate ∣∣∣∣ 1tn − 1T ∣∣∣∣ .
You know ∣∣∣∣ 1tn − 1T ∣∣∣∣ = |tn −T||tn||T| ,
think about what to do next.)