Can anyone think of an example of a sequence of functions $\displaystyle f_n:[0,1]\to\mathbb{R}$ that converges uniformly on $\displaystyle (0,1)$ but not on $\displaystyle [0,1]$?
If we don't assume $\displaystyle f_n$ need to be continuous a trivial example is:
$\displaystyle f_n(x)=\begin{Bmatrix} 0 & \mbox{ if }& x\in [0,1)\\(-1)^n & \mbox{if}& x=1\end{matrix}$
Fernando Revilla
can someone help me , i now what does mean the sequence of real number , but i confused what does it mean by sequence of vector space? for example vector space of polynomials ? and n*m matrices ?how would be their sentences of the related sequences? and why the convergent of these sequences depends on the norm that we define for these spaces ?