Consider the space of continuous, real-valued functions on $[0,1]$ under the sup-norm metric. I believe that the set of functions {$x^n$} for all natural n is a closed and bounded subset of our metric space that is not compact. It is bounded as the difference between any two functions in the set is at most $1$ and I believe it is not compact as it does not contain a convergent subsequence (no subsequence converges uniformly to a continuous function, let alone a continuous function in the set). I don't have a rigorous proof of closure but do believe it is closed. Hence, I believe that it works as an example of a subset of a metric space that is closed and bounded but not compact. Is my reasoning valid?
[Math] Is the set {$x^n$} a closed and bounded subset of C $[0,1]$ that is NOT compact
functional-analysisreal-analysis
Best Answer
Suppose it is not closed. Its complementary $U$ is not open. There exists $f\in U$ such that every ball $B(f,1/n)$ contains a function $x^m$. We deduce that $f=lim_nx^{u(n)}$. Since $f$ is not of the form $x^m$. $lim_nu(n)=+\infty$. But In this case, $lim_nx^{u(n)}$ is the function $g:[0,1]\rightarrow R$ such that $g(x)=0, x\neq 0$ and $g(1)=1$. Contradiction since $g$ is not continue, and thus cannot be equal to $f$.