Real Analysis - How to approach this question?

Question

I am currently taking Real Analysis and it is working out quite good. However, I feel like I can't comprehend the level of abstraction of exercises like this:

Let $M$ be an uncountable subset of the real numbers. Show that there is a convergent sequence whose terms are all distinct and lie in $M$.

How do I approach problems like this? In general, I seem to have some problems with sequences. When the sequence is given and I'm asked about proving the convergence, it's pretty straightforward, but given no terms and doing epsilon-delta with a general sequence is much harder.

Thank you and stay safe.

Hint: start by showing that there must be an integer $n$ such that uncountably many elements of $M$ are in the compact interval $[n, n+1]$. — lulu, Nov 26 '23 at 10:38
We can prove something better. Have a look at this: An uncountable subset of $\mathbb R$ has uncountably many limits points — Nothing special, Nov 26 '23 at 10:43
May I try to use a proof by contradiction? Apart from that, I don’t see how that helps. — complexlogarithm, Nov 26 '23 at 16:36
What you may do is go back and look at what you know about a set being "uncountable", and what you know about countable sets. Then look again at lulu's hint. How many intervals $[n,n+1]$ for integer $n$ are there? Suppose the intersection of each of them with $M$ was countable. What would that imply about $M$? Maybe that same idea could be applied further,,,, — Paul Sinclair, Nov 27 '23 at 05:41

Real Analysis - How to approach this question?

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