Let r > 0 be a positive real number. This problem will give a simple approach to lim n →∞ r n . i. Explain briefly why a n = r n can be written recursively as a 0 = 1, a n = r ⋅ a n -1 for n > 0. ii....

Letr > 0 be a positive real number. This problem will give a simple approach to lim
_n→∞r

ⁿ
.
i. Explain briefly whya

_n
 =r

ⁿ
 can be written recursively asa
₀ = 1,a

_n
 =r⋅a

_n-1 forn > 0.
ii. Using the technique of solving for the limit of a recursive sequence, find the possible limits ofa

_n
 if the sequence converges.
iii. Show thata

_n
 is monotone. (It may be increasing or decreasing, depending onr.)
iv. Combine (ii) and (iii) to show lim
_n→∞r

ⁿ
 converges to 0 forr < 1,="" converges="" to="" 1="">r = 1, and diverges to ∞ forr > 1.