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**Enrique2** Mean value theorem shows that B is equicontinuous .

Moreover, since $\displaystyle f(0)=0$, again mean value shows that, for $\displaystyle x\in (0,1)$

$\displaystyle |f(x)|=|f(x)-f(0)|\leq |x-0|=|x|$ for each $\displaystyle f\in B$. This yields that B is pointwise bounded (for x=1 is an easy consequence). Now Arzela Ascoli tells us that B is relatively compact, that is equivalent to the desired asertion.

By the way, the part of equicontinuity that I only mention above is better explained in Jose27 post