I am stuck in this question (image link given above). I have solved the first part that fn(x) converges to a continuous function but I am not getting any idea to solve the 2nd part to prove fn(x) doesn't converge uniformly. Please help me... I will post my work proving the first part in the comments.
Let $\epsilon =1/2$ , take $x_n=\frac 2{2n+1}$
Then show $1/(n+1) \le x_n \le 1/n$
Clearly $|f_n(x_n)|=1 \gt \epsilon$
Can you conclude ?