# Math Help - Limit proof

1. ## Limit proof

Given that (x sub n) is a sequence of real numbers, that x > 0, and that (x sub n) approaches x as n approaches infinity, prove that there exists an integer N such that the inequality x > 0 holds for all integers n greater than or equal to N.

2. Originally Posted by Slazenger3
Given that (x sub n) is a sequence of real numbers, that x > 0, and that (x sub n) approaches x as n approaches infinity, prove that there exists an integer N such that the inequality x > 0 holds for all integers n greater than or equal to N.
Let $\frac{x}{2}=\varepsilon>0$. By $x_n\to x$ there exists some $N\in\mathbb{N}$ such that $N\leqslant n\implies |x_n-x|<\varepsilon\implies x-\varepsilon=\frac{x}{2}