Prove that for every $n\in \mathbb{N}$, $n\ge 2$, and for every $1\le i\le\lfloor{\frac{n}{2}}\rfloor$: $$\binom{n}{i-1} < \binom{n}{i}$$ This is a homework problem and I suppose proof should be done by induction. I'm not quite sure what to do with this $i$ because it is not fixed.
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$$i \le \left\lfloor \frac n2\right\rfloor \le\frac n2 < \frac{n+1}2$$
– peterwhy Nov 02 '22 at 23:12