Why is the inequality changed from $<$ to $\le$ when taking the limit in the case below

Question

I am reading a proof in which it is true that for a sequence $(x_n)$, $$d(x_n, x_N) < \frac\epsilon 2 \quad \text{for all} \,n > N.$$

It is then said that it follows that $$\lim_{n \to \infty} d(x_n, x_N) \le \frac \epsilon 2.$$

My question is why is the 'sharpness' of the inequality changed?

Answer 1

Because, for instance, $(\forall n\in\mathbb{N}):1-\frac1n<1$, but $\lim_{n\to\infty}1-\frac1n=1$. In general, limits don't preserve strict inequalities.