Upper bound of the probability of a poisson random variable less than its mean?

Question

Considering $X\sim Poisson (\lambda)$, I am interested in establishing an upper bound for $P(X< \lfloor \lambda \rfloor)$ where $\lambda>1$. The following expression denotes this probability: $$ P(X< \lfloor \lambda \rfloor)=\sum_{k=0}^{\lfloor \lambda\rfloor-1} \frac{e^{-\lambda}\lambda^k}{k!} $$ Numerical tests suggest this probability is consistently under 0.5 and appears to converge to 0.5 as $\lambda$ tends towards infinity. I was wondering if there is a formal way to show that 0.5 is indeed an upper bound?