I need to prove the following:
Suppose that a random variable has a binomial distribution. Prove that if: $$0\leq k<k+1\leq E(X)$$ then: $$P(X=k)<P(X=k+1)$$
Essentially, I prove the inverse, that if $P(X=k)<P(X=k+1)$, then $k+1\leq E(X)=np$. This follows fairly simply from the definitions of $P(X=k)$ and $P(X=k+1)$ when you expand them into "binomial form" and do some cancellations. My question, therefore, is the following: is it legitimate to prove a statement in one direction, then argue that you could solve it in the opposite way by inverting the process you used to get there? In this particular example, all I really did was divide by common factors on both sides of the inequality, so I don't see why this type of argument cannot be made. However, I'm still skeptical of its validity, and perhaps curious about whether or not it's more generally applicable (from the precursory research I've done on the topic, it seems that it is not). Any information regarding this type of proof methodology would be appreciated. Cheers.