I am currently reading A course in point set topology by John B. Conway. The definition he give is standard: A sequence $\{x_n\}$ in X converges to X if for all $\epsilon>0$, there is an integer N such that $d(x,x_n)<\epsilon$ when $n\geq N$.
However, in remark, he says that we might also mention that the inequality $d(x,x_n)<\epsilon$ can easily be replaced by $d(x,x_n)\leq \epsilon$. Cleary the original definition implies the one in remark, but why does the backward implication hold?
