possible error while proving $\sum_{n=1}^{\infty}(x^{n} - x^{n-1})$ uniformly convergent in $[0,1)$ when its isn't

Question

possible duplicate of Why is $\sum_{n=1}^{\infty} x^n-x^{n+1} $ not uniformly convergent in $[0,1)$?

I'm trying to understand where exactly I went wrong trying prove this:

"Prove/refute: $\sum_{n=1}^{\infty}(x^{n} - x^{n-1})$ uniromly convergent over $[0,1)$"

So by Cauchy criterion a series is uniformly convergent iff for every $\epsilon>0$ there exist $N$ such thart for every $m>n>N$ and for every $x\in I$ takes place that $|(\sum_{k=n+1}^{m}f_{k}(x)|<\epsilon$

So I did this: Let $0<\epsilon$. There exists $N \in \mathbb{N}$ such that for every $N<n<m$ and for every $x \in [0,1)$:

$|\sum_{k=1}^{m}(x^{k} - x^{k+1}) - \sum_{k=1}^{n}(x^{k} - x^{k+1})|$

$<|\sum_{k=n+1}^{m}x^{k}(1-x)| <_{0\leq x<1} |\sum_{k=n+1}^{m}(1-x)|$

$ < |(m-n)(1-x)| = |m-n+ x(n-m)| $

$<_{0\leq x<1} |m-n+1(n-m)|$

$ = m-n+n-m = 0 < \epsilon$

So on one hand it seems like it is correct while here and also in my study books its not but I couldn't find where exactly I gone wrong.

Answer 1

You cannot possibly prove that $|m-n+x(n-m)|<0$, since no absolute value is negative. Your error lies in the assumption that $|m-n+x(n-m)|$ is smaller than the value that it takes when $x=1$.