Problem 45 (difficulty: 3/10)

Prove that the following identities hols for all positive integer \(\displaystyle n\):
 (1)  \(\displaystyle \displaystyle 1-\frac{1}{2} +\frac{1}{3} -\ldots -\frac{1}{2n} =\frac{1}{n+1} + \ldots +\frac{1}{2n};\)
 (2)  \(\displaystyle \displaystyle \frac{1}{1\cdot 2} +\ldots + \frac{1}{(n-1)\cdot n}=\frac{n-1}{n} .\)

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