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**serious331** Suppose that the function$\displaystyle f:R \rightarrow R $ is differentiable and that there is a bounded sequence $\displaystyle \{x_n\}$ with $\displaystyle {x_n} \neq {x_m}$ if $\displaystyle {n \neq m}$, such that $\displaystyle f(x_n) =0$ for every index n. Show that there is a point $\displaystyle x_0$ at which $\displaystyle f(x_0) = 0$ and $\displaystyle f'(x_0) = 0 $.

---> According to Bolzano–Weierstrass theorem, a bounded sequence has a convergent subsequence, but How can this be used to prove the above question.