If $S$ is an uncountable subset of $\mathbb{R},$ then does there always exists a strictly decreasing sequence $ (s_n) \in S,$ such that it converges to some point of $\mathbb{R}$
I divided $\mathbb{R}$ into union of closed sets $[n,n+1] n\in \mathbb{N}$ and atleast one such interval must have uncountable number of points. Hence it has a limit point, now how do I show the existence of a strictly decreasing sequence that converges to some point of $\mathbb{R}$?
Consider the set of points $s \in S$ such that $(s,s+\epsilon) \cap S$ is empty for some $\epsilon >0$. For each such $s$ we can consider a maximal interval $(s,s+\epsilon)$ with this property. These intervals are pairwise disjoint. It follows that the set of such points is at most countable. Now remove these ponits from $S$. Call this uncountable set $T$. There exists $n$ such that $T \cap [n,n+1]$ is uncountable. Now consider the infimum of the set $T \cap [n,n+1]$. Can you see that there is sequence of distinct points of $T$ which is decreasing to the infimum?