Answer :
Final answer:
If a sequence of numbers converges to two different limits, it contradicts the definition of a limit and the fact that there can only be one limit, proving the uniqueness of the limit.
Explanation:
The proof relies on the fact that in mathematics, limits are unique. That is, if a sequence {a_n} has a limit, then this limit is unique. We start by assuming the contrary, that there are two numbers L₁ and L₂ such that a sequence {a_n} converges to both L₁ and L₂. However, since {a_n} converges to L₁, for any positive number ε, after a certain term of the sequence, all the terms should lie in the interval (L₁-ε,L₁+ε). The same goes for L₂, all terms should lie in the interval (L₂-ε, L₂+ε). However, if L₁ ≠ L₂, we can choose ε to be less than 1/2|L₁-L₂|. Now, the intervals (L₁-ε,L₁+ε) and (L₂-ε, L₂+ε) can't intersect, which contradicts the assumption that there exist terms of the sequence both in (L₁-ε,L₁+ε) and (L₂-ε, L₂+ε) after a certain term of the sequence. Therefore, our original assumption that L₁ ≠ L₂ is incorrect. This means L₁=L₂, proving the uniqueness of the limit.
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