n-th Root Sequence

Proof:

Case a > 1:

If a > 1, then for n large enough we have 1 < a < n. Taking roots on both sides we obtain

• 1 < <

But the right-hand side approaches 1 as n goes to infinity by our statement of the root-n sequence. Then the sequence {} must also approach 1, being squeezed between 1 on both sides.

Case 0 < a < 1:

If 0 < a < 1, then (1/a) > 1. Using the first part of this proof, the reciprocal of the sequence {} must converge to one, which implies the same for the original sequence.

Case a = 0:

In this case we are dealing with the constant sequence, and the limit is of course equal to 1.