<s>[INST] Writing $1$ in form of $\frac{1}{t_1}+\cdots+\frac{1}{t_n}$ 
Possible Duplicate:
Prove that any rational can be expressed in the form $\sum\limits_{k=1}^n{\frac{1}{a_k}}$, $a_k\in\mathbb N^*$ 

Can anyone help me with this problem? It's a little strange:
Let $M$ be a natural number. Prove that we can write $1=\frac{1}{t_1}+\cdots+\frac{1}{t_n}$ such that all $t_i$'s are distinct natural numbers greater than $M$.
 [/INST] Hint. $$\frac{1}{n} +\frac{1}{n} = \frac{1}{n} + \frac{1}{n+1}+\frac{1}{n(n+1)}.$$
For example, say $N=3$. Then we can write:
$$\begin{align*}
1 &= \frac{1}{4}+\frac{1}{4} + \frac{1}{4}+\frac{1}{4}\\
&= \frac{1}{4} + \frac{1}{5}+\frac{1}{20} + \frac{1}{5}+\frac{1}{20}+\frac{1}{5}+\frac{1}{20}\\
&= \frac{1}{4}+\frac{1}{5}+\frac{1}{20} + \frac{1}{6}+\frac{1}{30} + \frac{1}{21}+\frac{1}{420} + \frac{1}{6}+\frac{1}{30} + \frac{1}{21}+\frac{1}{420}\\
&= \frac{1}{4}+\frac{1}{5}+\frac{1}{20}+\frac{1}{6}+\frac{1}{30}+\frac{1}{21}+\frac{1}{420} + \frac{1}{7}+\frac{1}{42} + \frac{1}{31}\\
&\qquad\mathop{+}\frac{1}{930} + \frac{1}{22}+\frac{1}{462} + \frac{1}{421}+\frac{1}{176820}\\
&= \frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{20}+\frac{1}{21}+\frac{1}{22}+\frac{1}{30}+\frac{1}{31}+\frac{1}{42}\\
&\qquad\mathop{+}\frac{1}{420}+\frac{1}{421}+\frac{1}{462}+\frac{1}{930}+\frac{1}{176820}.
\end{align*}$$
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