I'm trying to use the principle of mathematical induction to prove that $n!+n\ge2^n$ for all $n\ge1$. I know how to prove $n!\ge2^n$, but I can't figure out what to do with the extra $n$. For instance, if I try to factor $n+1$ out of the inductive case,
I'm left with $n!+1$, which I don't know what to do with. Similarly, if I try to use multiplicative telescopy, which I used to prove $n!\ge2^n$, I get stopped by the random $n$ that I can't multiply.
Any help would be greatly appreciated.