Fifteen squirrels gather $100$ nuts. Prove that two of the squirrels gathered the same number of nuts.
I am fully aware that this is a very elementary problem but I just need to make sure my method of proof is acceptable.
So this is what my proof was:
If each squirrel gathered a unique number of nuts, then:
$1 + 2 +\cdots+15 = 120$
$120$ is greater than $105$, therefore, at least two squirrels gathered the same number of nuts.
However, the proof in the solutions manual was as follows:
For each $i = 1, 2, \ldots , 15,$ let $x_i$ be the number of nuts gathered by squirrel $i$. Sort the $x_i$’s so that $x_1 ≤ x_2 ≤ · · · ≤ x_{15}$. If all the inequalities are strict, then $x_i ≥ i − 1$ for all $i$, and therefore $\sum x_i ≥ 105$.
The above proof seems to be quite technical though it does make sense. But I just wanted to know if my proof is sufficient or is the level or technicality in the official solutions the only acceptable method?