<s>[INST] Find the sum of the following series: $ \frac{1}{1 \cdot 2}\,+\,\frac{1}{2 \cdot 3}\,+\,\frac{1}{3 \cdot 4} +\cdots\,+\frac{1}{100 \cdot 101}$ 
Find the sum of the following series
  $$ \frac{1}{1 \cdot 2}\,+\,\frac{1}{2 \cdot 3}\,+\,\frac{1}{3 \cdot 4} +\cdots\,+\frac{1}{100 \cdot 101}$$

My Attempt:
$$\frac{1}{2 \cdot 3} =\frac{1}{2}-\frac{1}{3}.$$
So we can write question as:
$$\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\cdots+\frac{1}{99}-\frac{1}{100}+\frac{1}{100}-\frac{1}{101}\\
=\frac{1}{1}-\frac{1}{101}\\
=\frac{100}{101}.$$
Am i right?
 [/INST] _Sum up of everybody else's comments: _Yes, it is correct. The only thing I'd say is that as Theophile said, you need to make your prove more convincing, and back up every single hole in. A proof is never at an optimal proving state, but you can maximise this, by filling in every single statement you make with a law or rule, or in other cases, another statement. In the end, all the loose bits would be tied up, and your proof will be complete. 
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