Sequence of Binomial Coefficients is Strictly Increasing to Half Upper Index

Theorem
Let $n \in \Z_{>0}$ be a strictly positive integer.

Let $\dbinom n k$ be the binomial coefficient of $n$ over $k$ for a positive integer $k \in \Z_{\ge 0}$.

Let $S_n = \left\langle{x_k}\right\rangle$ be the sequence defined as:
 * $x_k = \dbinom n k$

Then $S_n$ is strictly increasing exactly where $0 \le k < \dfrac n 2$.

Proof
When $k \ge 0$, we have:

In order for $S_n$ to be strictly increasing, it is necessary for $\dfrac {n - k} {k + 1} > 1$.

So:

Thus $\dbinom n {k + 1} > \dbinom n k$ $k + 1$ is less than half of $n$.

Hence the result.