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Understand the problem

>True or false: Let $X \subseteq \Bbb Q^2$. Suppose each continuous function $f:X \to \Bbb R^2$ is bounded. Then $X$ is finite.
Source of the problem
TIFR GS 2019, Part B Problem 3
General topology
Medium
Suggested Book
Topology, Munkres

Do you really need a hint? Try it first!

Use the fact that every convergent sequence is bounded. So can you take a convergent sequence as a subset of $$\Bbb Q^2$$ and name it as $$X$$?
Consider $$X = \{(1,0), (1/2,0), (1/3,0), …, (0,0)\}$$. What can we say about the behaviour of $$f(x)$$ as $$x\to (0,0)$$?
As $$f$$ is continuous so $$f(x) \to f(0,0)$$. Hence $$f(X)$$ is convergent sequence. Now you are almost there!!
So, $$f$$ is bounded by Hint 1. Hence we satisfy all the hypothesis of the question that any function on $$X$$ is bounded but $$X$$ is infinite.

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