I have to prove $n<n!$ for all $n>2$ by mathematical induction. 

I did it as follows. I proved the base case. 

Then let it be true for $K>2$:

$$ K<K! $$

I have to prove,

$$ K+1<(K+1)! $$

$$ K<K! $$

Adding $1$ on both sides

$$ K+1<K!+1<(K+1)! $$

Hence

$$ K+1<(K+1)! $$

Is the last step I did ($K!+1<(K+1)!$) Right? 

Please help me out. Thanks!