Analysis with continuous, sum.

Hello teacher~

f(x) = sum{n=1 to oo} (x^n)(1-x) / n

Show that f(x) is continuous on [0, 1].

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Hm...maybe,
I must show that f(x) converges uniformly on [0, 1] ?

Hm...
For n >= m > N,
|f_n(x) - f_m(x)| =
|{(x^(m+1))(1-x) / (m+1)} + ... + {(x^n)(1-x) / n}|
<= (1-x).|{x^(m+1) / (m+1)} + ... + {(x^n) / (m+1)}|
<= (n-m)(x^(m+1)) / (m+1)
<= (n-m) / (m+1)
??

Maybe, I need your rebuke.


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