Let $X$ be a reduced projective curve over a field $k$ with irreducible components $X_1,\ldots,X_r$. Let $p_a(X) = 1- \chi(\mathcal{O}_X)$ denote the arithmetic genus of $X$. Moreover, denote by $X_i'$ the normalization of $X_i$.
Do we know that $p_a(X_i') \leq p_a(X)$?
Background:
I would like to obtain a bound on the number $r$ of irreducible components in terms of $p_a(X)$. And to obtain such a bound I would like to use Proposition 7.5.4 in Liu's book which says
$$p_a(X)+r-1 = \sum_{1\leq i \leq r} p_a(X_i') + \sum_{x\in X} \dim_k \mathcal{O}_{X,x}'/\mathcal{O}_{X,x}$$
where $\mathcal{O}_{X,x}'$ denotes the integral closure of $\mathcal{O}_{X,x}$ in its total ring of fractions.