Let X be a metric space and let (s_{n}
)_{n} be a sequence whose terms are in X. We say that (s_{n}
)_{n} converges to s∋X if
∀ϵ> 0∃N∀n ≥ N : d(s_{n},s) <ϵ
For n ≥ 1, let j_{n} = 2[(5^(n) - 5^(n-1))/4].
(Convince yourself that 5^(n) - 5^(n-1) is always divisible by
4, so the exponent in the definition is always a positive integer.) The first few terms of this
sequence are
2; 32; 33554432; 42535295865117307932921825928971026432
so you would reasonably expect this sequence to diverge with respect to the usual metric on
Q (the one given by the usual absolute value).
However, show that |j^{2}_{n} - (-1)|_{5} ≤ 5^(-n) where ||_{5} is the 5-adic absolute value.
My Attempt:
I started by writing the claim in terms of v_{5}(j^{2}_{n} + 1). Then i tried to find a recurrence that looks like this:
(j^{2}_{n}+ 1)^5 = (^{2}_{n+1}+1) + (some other stuff).
I was thinking I can show that the sequence (j_{n})_{n} is also Cauchy with respect to ||_{5}, so in Q_{5},the completion of Q with respect to ||_{5}, the sequence (j_{n})_{n} converges to a number j∋Q5 such that j^{2} = -1. It follows that Q5 is not an ordered field, unlike the completion of Q with respect to the usual ||_{5}, which is our old friend R.