Prove that $\sum_{n=1}^{\infty} \frac{n}{(n+1)!}$ converges to 1

Question

Proving the convergence can be easily done using the quotient rule.

But I'd like to prove that this series actually converges to 1.

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I was thinking of proving that:

• $\sum_{n=1}^m \frac{1}{2^n} \leq \sum_{n = 1}^m \frac{n}{(n+1)!}$
• $\sum_{n = 1}^m \frac{n}{(n+1)!} \leq 1$

and then applying the squeeze theorem.

I tried to prove those statements by induction, but I got stuck.

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Are there simpler ways to tackle this problem?