Consider the sequence of continuous functions \(f_n: \mathbb{R} \to \mathbb{R}\) defined by
\[ f_n(x) = x/n. \]
In which of the three topologies of Theorem 46.7 does this sequence converge? Answer the same question for the sequence given in E-BTI7W.
Solution
Theorem 46.7 compares pointwise, compact-convergence, and uniform topologies.
For \(f_n(x)=x/n\), we have pointwise convergence to \(0\). On every compact \(C\subset\mathbb R\), \(M=\sup_{x\in C}|x|<\infty\), so \[ \sup_C|f_n|=M/n\to0; \] hence convergence is compact-uniform. It is not uniform on all of \(\mathbb R\), since for every \(n\), \(\sup_x|x|/n=\infty\).
For the sequence of E-BTI7W, \[ f_n(x)=\frac1{n^3(x-1/n)^2+1}, \] \(f_n(x)\to0\) pointwise by E-BTI7W. It does not converge uniformly because \(f_n(1/n)=1\). Nor does it converge in the compact-convergence topology: the compact set \([0,1]\) contains \(1/n\) for all \(n\), so \[ \sup_{[0,1]}f_n=1. \] Thus the first sequence converges pointwise and compact-uniformly but not uniformly; the second converges only pointwise.