In the book introduction to real analysis by Robert G. Bartle i came across a problem
Give an example of a decreasing sequence $(f_n)$ of continuous functions on $[0,1)$ that converges to a continuous function, but the convergence is not uniform.
I thought an example as $f_n(x)=1/(n(1-x))$ is a decreasing sequence of continuous functions on $[0,1)$ and $limf_n(x)=0$ for all x $ \in [0,1)$ is continuous function.(is this pointwise limit is right?)
Now let a sequence $n_k=k$ and sebsequence $x_k=k/k+1$ in $[0,1)$ then $\vert f_{n_k}(x_k)-f(x_k) \vert =1/k(1-k/k+1)=k+1/k=1+1/k \ge 1$ for all $k\in \Bbb N$
so convergence is not uniform. I s this is okk??
I think this example is ok. And the pointwise limit is also right. I think the following ideas will help you: When we investigate the pointwise limit we just choose any number in a set, then fix it and this means we just think x is constant and let n tend to infinity ; When we study on the uniform convergence we don't fix x, it means maybe the point which is not uniform converge is changing, so at the first time we make n fixed and find when $|fn(x)-f（x）|$ gets maximum for every n, and in this process x in fact depends on n. Another way to testiy if the $sup|fn（x）-f（x）|$ tend to 0 when n tend to infinity. In this example, when $x＝1$ this value get its supreme $1/n$（though we can't get 1,but the supreme can be sure ）,and this satisfies.