Determine whether the following sequence converges pointwise or unifomly Determine whether the following sequences converge pointwise or unifomly on the
given set.
$f_n(x)=\frac{x+n}{\:n}$ on $[a,b]$
I know that pointwise limit is 1.
Thank you

Let $f_n(x)=1+\frac xn$.  Then, we have for any $\epsilon>0$
$$\begin{align}
\left|f_n(x)-1\right|&=\frac{|x|}{n}\\\\
&\le \frac{\max\left(|a|,|b|\right)}{n}\\\\
&<\epsilon
\end{align}$$
whenever $n>\frac{\max\left(|a|,|b|\right)}{\epsilon}$.  Hence, $f_n(x)$ converges uniformly to $1$ for $x\in [a,b]$.

The convergence fails to be uniform for $x\in \mathbb{R}$.   Take $\epsilon=1$.  Then, for all $N$ we have for $x=n$ and $n=N+1$
$$\begin{align}
\left|f_n(x)-1\right|&=\frac{|x|}{n}\\\\
&=1\\\\
&\ge \epsilon
\end{align}$$
