For random variable $X$: $\mathbb{E}|X| < \infty$. Does it imply that for every $a \in \mathbb{R}$ there exists $\mathbb{E} | X - a | < \infty$?

Question

For random variable $X$ we know that: $\mathbb{E}|X| < \infty$ does it imply that for every $a \in \mathbb{R}$ there exists $\mathbb{E} | X - a | < \infty$?

Edit:

In general we know that: $\ \mathbb{E}|X−a| \leq \mathbb{E}(|X|+|a|)$

From linearity of expectation we know that: $\ \mathbb{E}|X|+\mathbb{E}|a|$

Therefore: $\mathbb{E}|X−a| \leq \mathbb{E}|X| + \mathbb{E}|a|$ and from here:

• form assumption we know that: $\mathbb{E}|X| < \infty$
• in general we know that: $\mathbb{E}|a|=|a|$ and $ |a| < \infty$ for $a \in \mathbb{R}$

I used help of @geetha290krm @Michael. Is that correct?