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$$L =\left \{ \langle M \rangle \mid \exists x\in \Sigma^* \left(\left | x \right |\leq 10000 \wedge H(M, x\right) \right \}$$

Where $H(M, x)$ denotes whether Turing machine $M$ halts on input $x$.

My guess is neither $R$ or $Re$ because I would have to run infinite TM's in order to find out if every TM halts on x or is not halting on x. Is my intuition correct?

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The language is in RE: simply generate all inputs $x$ with $|x|\le 10000$ (they are finitely many), and simulate $T$ on all inputs in parallel. Whenever one simulation halts, accept.

The language is not in co-RE, as otherwise it would be decidable and we could solve the Halting problem. Indeed, to decide whether a Turing machine $T$ halts on empty input, we could create a new machine $M$ that simulates $T$ if $x=\varepsilon$ and does not halt otherwise. Then $M \in L$ if an only if $T(\varepsilon)$ halts.

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