A moving peak and uniform error
A fictional analysis exercise studies f_n(x)=nx/(1+n²x²)on[0,1]. The group first fixes x and examines the limit, then compares the result with a supremum-norm requirement. A learner samples only the endpoints and concludes that convergence is uniform. Another notices a peak whose location depends on n but does not yet know whether that invalidates pointwise convergence. Students must keep the quantifiers separate and use an explicit point sequence to evaluate the largest error. They should also explain why restricting the domain to[a,1]with a fixed positive a changes the available bound. No numerical plot alone is accepted as proof of an assertion about every n and every point.
1. What happens pointwise and uniformly on[0,1]?
Read the worked answer
For each fixed x, the sequence tends to zero. At x=1/nits value is1/2, so the supremum error does not tend to zero and convergence is not uniform.
2. Why does[a,1]behave differently?
Read the worked answer
For x≥a>0, f_n(x)≤1/(nx)≤1/(na), which tends to zero uniformly. The fixed positive lower bound excludes the moving peak near zero.