Using induction to prove that 2^x =4

Question

Using induction to prove that 2^x < x! for all x>=4

anonymous · Answer

you know how to do a proof by induction?

anonymous · Answer

if so, after the base case this takes one step
if not, then it takes the rest of the night to explain it

solomonzelman · Answer

why not for all x≥2?

anonymous · Answer

cause $$2^2>2!$$

solomonzelman · Answer

oh yes, I read the equation the other way around

solomonzelman · Answer

factorial is greater!

solomonzelman · Answer

well, you can tell that

$\large\color{#000000 }{ \displaystyle 2^4=16 }$        $\large\color{#000000 }{ \displaystyle 4!=24 }$ 

multiply          mutliply
× 2                 × larger and larger number
every time       every time

anonymous · Answer

is totally obvious, but you have to understand the mechanics of a proof by induction
if you know what they are, this one is solved in one step as @SolomonZelman wrote above
if not, then it takes for ever

solomonzelman · Answer

I recalled this logic from the very first proof of Harmonic Series Divergence - b/c that's the first time I've seen this....

I mean that:
1/5+1/6+1/7+1/8
is at least
1/8+1/8+1/8+1/8

and so on...

solomonzelman · Answer

you are multiplying times number that is at least 2, for x!