formalize the following statment: a_n is a sequence that doesnt converge to L∈R. prove that the sequance (1/n) does not converge to 1.
Can someone please explain to me what i have to do in the question i do not understand it so well.
To formalize the statement "a_n is a sequence that doesn't converge to L ∈ R," we can use mathematical notation. Let's denote the sequence as (a_n), and we can write the formal statement as follows:
(∀ L ∈ R) (∃ ε > 0) (∀ N ∈ N) (∃ n > N) (|a_n - L| ≥ ε)
This statement essentially says that for any real number L, there exists some positive value ε such that for any natural number N, there exists an index n greater than N where the absolute difference between a_n and L is greater than or equal to ε.
Now, we want to prove that the sequence (1/n) does not converge to 1. To do this, we can show the negation of the definition of convergence. In other words, if we can find an ε such that for any N, there exists n greater than N where |1/n - 1| ≥ ε, then we can conclude that the sequence does not converge to 1.
Let's choose ε = 1/2. Now, for any natural number N, we need to find an index n greater than N such that |1/n - 1| ≥ 1/2. We can achieve this by considering n = 2N. In this case, |1/(2N) - 1| = 1/2. Since 1/2 ≥ ε = 1/2, we have successfully shown that for any N, there exists an index n greater than N where |1/n - 1| ≥ ε.
Therefore, by proving the negation of the definition of convergence, we have shown that the sequence (1/n) does not converge to 1.