Uniform convergence of 1 - (x/n)

Question

Is $1 - \frac{x}{n}$ uniformly convergent on $\mathbb{R}$ (set of reals)?

I am having a problem with finding the point-wise limit of the sequence for the domain $(-n,n)$. Please help!

Answer 1

If the sequence were uniformly convergent, we could find an $n$ such that $|1-\frac{x}{n} - 1| = \frac{|x|}{n} \leq \epsilon$ for all $x \in \mathbb{R}$. This is obviously not the case, for any given $n$, just consider $x=2n$.

Answer 2

It is easy to show that $\lim_{n\to \infty}\left(1-\frac xn\right)=1$. But for $\epsilon=\frac12$ and for all $N$, there exists a number $n>N$ and a number $x$, say $x=n$, such that

$$\left|1-\frac xn-1\right|=1 \ge\frac12=\epsilon$$

This negates the uniform convergence.