1. quick question about inductive proof

Prove that: $3^n < n!$ n is an integer greater than 6.

Basic step:
$P(7): 3^7 < 7!$
2187 < 5040

Inductive Steps:
hypothesis: $P(k): 3^k < k!$
conclusion: $P(k+1): 3^{k+1} < (k+1)!$

Proof: $3^(k+1) = 3.3^k < 3.k! < (k+1)k! = (k+1)!$

why is this one: $(k+1)k!$ equal to $(k+1)!$?

Is there any better way to prove it?

2. If you have more than one character in an exponent, set off the whole exponent in braces.
$$3^{(k+1)}$$ gives $3^{(k+1)}$ instead of $3^(k+1)$.

3. Originally Posted by zpwnchen
Prove that: $3^n < n!$ n is an integer greater than 6.

Basic step:
$P(7): 3^7 < 7!$
2187 < 5040

Inductive Steps:
hypothesis: $P(k): 3^k < k!$
conclusion: $P(k+1): 3^(k+1) < (k+1)!$

Proof: $3^(k+1) = 3.3^k < 3.k! < (k+1)k! = (k+1)!$

why is this one: $(k+1)k!$ equal to $(k+1)!$?

Is there any better way to prove it?
$k! = 1\cdot 2 \cdot 3 \cdot ... \cdot k$
$(k+1)! = 1\cdot 2 \cdot 3 \cdot ... \cdot k \cdot (k+1)$
$(k+1)k! = (k+1)(1\cdot 2 \cdot 3 \cdot ... \cdot k) = 1\cdot 2 \cdot 3 \cdot ... \cdot k \cdot (k+1) = (k+1)!$