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Proof for formula for sum of sequence $1+2+3+\ldots+n$?

I have a question:

Show that $1+2 + \cdots+ n = \binom{n+1}{2}$

This is a combinatorial proof. Here is my thinking:

Proof. ($\Leftarrow$): Suppose we have $n+1$ objects labeled $1$ to $n+1$. Then there are $n$ ways to pair $1$ with the other objects, $n-1$ ways to pair $2$ with the other objects (since $12$ is the same as $21$), etc.

Is that correct? How would we do the other direction?

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    About "the other direction": There's an "other direction" in proving equivalences, because an equivalence $\Leftrightarrow$ can be considered as the conjunction of two statements, $\Rightarrow$ and $\Leftarrow$. The analogue for an equality would be to consider it as a conjunction of two statements, $\ge$ and $\le$; then there would be two directions. However, that's not what you did, you proved (or started to prove) the equality itself, and when you do that, there's no other direction left to prove.2012-11-11

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