Strictly increasing chain of ideals in continuous functions on [0,1].

Question

Let $\mathcal{C}[0,1]$ be the ring of continuous functions on $[0,1]$. Is the following chain of ideals strictly increasing:

$(x)\subset (x,x^{1/2})\subset (x,x^{1/2},x^{1/3})\subset \cdots \subset (x,x^{1/2},\ldots ,x^{1/n})\subset\cdots$.

Answer 1

If $xf_1+x^{1/2}f_2+\cdots+x^{1/n}f_n=x^r$ for some $r$, $0<r<1/n$, then dividing through by $x^r$ gives a function on the left vanishing at zero, but on the right the constant $1$, contradiction.