Equivalent definition of convergence of a sequence in $\mathbb{R}$

Question

While reading over the definition of a Cauchy sequence in $\mathbb{R}$, I conjectured that the following condition

$\forall \epsilon>0, \exists N\in\mathbb{N}: \forall n>N, |a_n-a_N|<\epsilon$

is necessary and sufficient for a sequence in $\mathbb{R}$ to converge.

Is this true? I haven't been able to find any counterexamples or proofs of this conjecture on this site.

If this is indeed true, I'm guessing that a proof would involve subsequences and/or the Cauchy sequence condition.

Answer 1

You don't even have to discuss convergence - the condition you specify is equivalent to being a Cauchy sequence in any metric space. Just use the triangle inequality

$$|x_m-x_n|\le |x_m -x_N|+|x_n-x_N|$$

and realize that if you can make both of the terms on the right hand side small then you can make the term on the left hand side small.

(You obviously should make the argument more formal. If you've seen a standard $\epsilon/2$ argument before it won't be hard. And if you don't recognize such an argument, here's some good practice!)