Subset of infinite connected set

Question

How to proove that infinite connected set has got proper infinite connected subset?

Answer 1

In this question and its answer, it is shown that a connected space $X$ has at most one dispersion point. This is a point $p \in X$ such that $X \setminus \{p\}$ (in its subspace topology) doesn't have any connected subsets except singletons.

Now if $X$ is infinite and $T_1$ (points are closed; I see the tag metric-spaces, so in such spaces this is certainly the case), consider a point $p$. If $X \setminus \{p\}$ is connected, we are done. If $X \setminus \{p\}$ has an infinite connected set, we are done as well, so we assume that $X \setminus \{p\}$ only has finite connected sets. But finite sets are never connected in a $T_1$ space (they are discrete). So in fact, for a point $p$, there are two options: it is "good" in the sense that $X\setminus \{p\}$ has an infinite connected subset, or it is a dispersion point of $X$.

As said, there can be at most one dispersion point, so all other points of $X$ are "good": and so we are done.