Using Markov's Inequality to Derive a Conclusion about random variable

Question

I'm wondering whether I can use Markov's inequality to reach the following statement:

Given Markov's inequality on a non-negative random variable X:

$ P[X\geq a] \leq \frac{E[X]}{a}$

We can do the following:

$ P[X<a] = 1 - P[X\geq a]$

Thus, we can say:

$ P[X<a] \geq 1 - \frac{E[X]}{a}$

Now, since we know that $ 0\leq P[X<a] \leq 1 $, we can conclude that:

$E[X] \leq a$

What do you guys think?

Thanks in advance for your thoughts and ideas.