Why are the (connected) components of a topological space themselves connected?

I am trying to prove that (connected) components of a topological space are connected. I’ll first define what I mean by a ‘component of a topological space’:

For a topological space $X$, write $x\sim y$ if $\exists\ Y \subset X$ such that $Y$ is connected and $x, y \in Y$ (this is an equivalence relation. The components of $X$ are the equivalence classes for $\sim$.

My proof so far of why components are connected:

Let $C$ be a component, $x_0 \in C$. Then $\forall y \in C, \exists A_y \subset X$ such that $A_y$ is connected, with $x_0, y \in A_y$. Note that $A_y \subset C, C = \bigcup_{y \in C}A_y$…

Note sure how to finish this off. Any insight would be appreciated!

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