1. The problem statement, all variables and given/known data
Prove that if ##0<c<1## then llim##c^{\frac{1}{n}}=1## using the monotone convergence theorem.
2. Relevant equations
3. The attempt at a solution
I let ##c_n=c^{\frac{1}{n}}## and it follows since ##0<c<1 \implies 0<c^{\frac{1}{n}}<1## Thus ##c_n## is bounded above by 1. Now I want to show that ##c_n## is monotonically increasing by induction but im not sure how to do it. So for my base case I know I need to show ##c_1<c_2## And for my inductive case I suppose that ##c_k<c_{k+1}## and show ##c_{k+1}<c_{k+2}## which is what im stuck on.
Prove that if ##0<c<1## then llim##c^{\frac{1}{n}}=1## using the monotone convergence theorem.
2. Relevant equations
3. The attempt at a solution
I let ##c_n=c^{\frac{1}{n}}## and it follows since ##0<c<1 \implies 0<c^{\frac{1}{n}}<1## Thus ##c_n## is bounded above by 1. Now I want to show that ##c_n## is monotonically increasing by induction but im not sure how to do it. So for my base case I know I need to show ##c_1<c_2## And for my inductive case I suppose that ##c_k<c_{k+1}## and show ##c_{k+1}<c_{k+2}## which is what im stuck on.
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