Proof of an equivalent statement for the convergence of a sequence

Question

Can someone provide me a hint to how I should go about proving this statement:

$2.$ Prove that a sequence $\{a_n\}$ (of real numbers) converges if and only if corresponding to an arbitrary positive number $\epsilon$ there exists a number $N$ such that $n>N$ implies $|a_n - a_N| < \epsilon$.

Thanks!

Answer 1

Hint: for one direction, use the inequality: $$ |a_m-a_n|<|a_m-a|+|a_n-a| $$

for the other one, one needs the completeness of the real numbers.