Does $y_n=\frac{nt}{1+nt}$ converge in the space $(C[0,1], d_\infty)$?

Question

What I have done so far is that for $t \in (0,1]$ this sequence converges to 1 pointwise. At $0$, it converges to $0$. Then, can I conclude from these that if this sequence were to be convergent, then the limit must be the function $f(t) = 0$, for $t=0$, and 1 otherwise? If yes, then this function is not continuous. Thus, this sequence is divergent in this space.

I am not sure about this argument, and I used a graphing program to graph these functions. It looks like it is a Cauchy sequence, and so since C[0,1] is complete, the sequence is convergent. Therefore, I am confused right now.

Can you explain whether it is convergent or not more precisely. Thanks for the help in advance.

Answer 1

No.

There are two different kinds of convergence involved here

• pointwise convergence,
• uniform convergence, namely, convergence in the complete metric space $X=(C([0,1]),d_\infty)$.

The sequence $y_n$ converges pointwise to the function $f$. But, as you have observed, the sequence does not converge uniformly (i.e. it does not converge in $X$) to $f$ since $f$ is not continuous.