I have that $\displaystyle 1+\sum_{i=1}^{2n} \dbinom{2n}{i} (-y)^i \geq d$, where $\displaystyle 0 \leq y <1$

Im trying to figure out why the following inequality holds:

For any $\displaystyle P \in \{2, 4, 6, 8, ..., 2n\}$

$\displaystyle 1+\sum_{i=1}^{P} \dbinom{2n}{i} (-y)^i \geq d$

Why is this??

I checked it emphirically for $\displaystyle n \leq 30$ and it always holds