>>12511851>>12511855Okay, but what if I have another sequence that also "converges" to "sqrt(2)"?
As a concrete example, define:
a_0 = 1
a_n = a_{n - 1} + k_n * 10^-n
Where k_n is the nth decimal digit. k_n can be found by just trying a_{n - 1} + {0,...,9} * 10^-n and determining the largest value for k_n that doesn't allow the expression's square to exceed 2. This is also a sequence and should intuitively also converge to sqrt(2) right?
Looks like you're going to need an equivalence relation. The standard one between two sequences is:
e(s_n, s_n') = [lim(n -> inf) (s_n - s_n') = 0]
Now how are you going to check whether your sequence and my sequence, or in general, any arbitrary sequence converges to the same value. Well, you're going to have to determine:
For all e > 0, there exists m, n >= N s.t. |s_m - s_n| < e
for the difference between the sequence you gave and any other sequence. Well, since sequences are modeled as choice functions, you have, in general, an infinite amount of work ahead of you (you're not always going to be able to express N in terms of e). You can't escape the intrinsically infinitary nature of the construction of the real numbers.
