Prove the Root Property for Limits: Suppose {a,} converges to a, a, 2 0 for eachnEN, and m E N. Prove that {anm} converges to a'/m. That is, prove that, if {an}is a sequence of non-negative terms, then for all m e N,lim an =/ lim ann-00provided lim an exists.00-u(HINTS:• First, argue that, under the hypotheses, the number a must be non-negative.• Prove the result first in the case that a = 0.• To prove the result if a > 0, use the Difference of Powers Formula:т-1m-1-k k1" – y" = (x – y)Ek=0anm and y == a'/m and the Comparison Lemma.You may use standard properties of exponents like (r") = ra$.)with r

Name: Prove the Root Property for Limits: Suppose {a,} converges to a, a, 2
Uploaded: 2024-03-20T06:38:58.000Z
Channel: Bartleby
Description: Prove the Root Property for Limits: Suppose {a,} converges to a, a, 2 0 for eachnEN, and m E N. Prove that {anm} converges to a'/m. That is, prove that, if {an}is a sequence of non-negative terms, then for all m e N,lim an =/ lim ann-00provided lim