Proof help...

[Imum Coeli]

Hi I've just started real analysis and I'm so lost. I was wondering if someone could please tell me if my proofs are total nonsense.

Question: Let \(\displaystyle (a_n)\) be a convergent sequence of integers with \(\displaystyle \lim_{n \to \infty} \ a_n = L \). Show that the sequence must eventually be constant.

Answer:
Since all convergent sequences are Cauchy then \(\displaystyle (a_n) \) is Cauchy.
Since \(\displaystyle a_n, a_m \in \mathbb{Z} \) then if \(\displaystyle \epsilon \leq 1 \) we must have \(\displaystyle a_n =a_m \; \forall n,m \geq N \) where \(\displaystyle N \in \mathbb{R} \) (by definition of Cauchy sequences)
But since \(\displaystyle (a_n)\) converges to \(\displaystyle L \; \exists N_1 \in \mathbb{R} : |a_n - L|<\epsilon \; \forall n \geq N_1 \)
Therefore if \(\displaystyle n \geq N_1 \) then \(\displaystyle a_m = L \; \forall m \geq N_1 \)
So \(\displaystyle a_n = L \; \forall n \geq N_1 \)

 

[stapel]

I'm not sure, but I think you've proven that, given the sequence is eventually constant, it must then converge to L. That's backwards, I think.

How about something like this: Start with the fact that the sequence converges to L. This means that, given \(\displaystyle \epsilon \, >\, 0,\) there is some N such that \(\displaystyle L\, -\, a_n\, <\, \epsilon\) for all n > N. Pick an \(\displaystyle \epsilon \, <\, 1\), and... chase the definitions from there. You should easily arrive at "and thus all further terms must equal L".

 

[Imum Coeli]

Thanks heaps but for some reason I'm fixed on using a Cauchy sequence.

What if I revise the same argument:

Since all convergent sequences are Cauchy then \(\displaystyle (a_n) \) is Cauchy.
Let \(\displaystyle \epsilon = 1/2 \) then by definition of Cauchy sequences \(\displaystyle \exists N : |a_n - a_m|<1/2 \; \forall n,m \geq N \)
but since \(\displaystyle a_n, a_m \in \mathbb{Z} \) then \(\displaystyle a_n =a_m \; \forall n,m \geq N \)
\(\displaystyle \therefore (a_n)\) is constant \(\displaystyle \forall n \geq N \)