(Solved): Please provide all work necessary in order to receive a thumbs up. a) Let \( f \) be a real-val ...

Please provide all work necessary in order to receive a thumbs up.

a) Let \( f \) be a real-valued continuous function on \( [0,1] \) and suppose \( f(0)= \) \( f(1) \). Prove that there exists a point \( x \in[0,1 / 2] \) such that \( f(x)=f(x+1 / 2) \). a) Let \( \left(a_{n}\right)_{n=1}^{\infty} \) and \( \left(b_{n}\right)_{n=1}^{\infty} \) be sequences of real numbers and suppose \( \sum_{n=1}^{\infty} a_{n}^{2} \) and \( \sum_{n=1}^{\infty} b_{n}^{2} \) both converge. Show that \[ \sum_{n=1}^{\infty}\left|a_{n} b_{n}\right| \] converges.