A Nice Algebra Problem (if such exists)

Algebra Level 3

\[\frac 1{1\cdot 2013} + \frac 1{2\cdot 2012} + \frac 1{3\cdot 2011} + \cdots + \frac 1{2012 \cdot 2} + \frac 1{2013\cdot 1} < 1 \]

Is the expression above true?

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