Determine whether the following sequence converges pointwise or unifomly

Question

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

Answer 1

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 $x\in [a,b]$.

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The convergence fails to be uniform for $x\in \mathbb{R}$. Take $\epsilon=1$. Then, for all $N$ we have for $x=n$ and $n=N+1$

$$\begin{align} \left|f_n(x)-1\right|&=\frac{|x|}{n}\\\\ &=1\\\\ &\ge \epsilon \end{align}$$