Math and science::Analysis::Tao::06. Limits of sequences

Uniqueness of convergence, proposition

Let \( (a_n)_{n=m}^{\infty} \) be a sequence of reals, and let \( L \ne L' \) be two distinct real numbers. Then \( (a_n)_{n=m}^{\infty} \) cannot converge to both \( L \) and \( L' \).

Proof

If \( L \ne L' \), then there exists a positive real \( d = dist(L, L') = | L - L'| \). As \( (a_n)_{n=m}^{\infty} \) converges to \( L \), we can choose a real \( 0 < \varepsilon < \frac{d}{3} \) and there exists an integer \( N_1 > m \) such that \( (a_n)_{n=N_1}^{\infty} \) is ε-close to \( L \). Similarly, for \( L' \) for the same \( \varepsilon \), there. exists an integer \( N_2 > m \) such that \( (a_n)_{n=N_2}^{\infty} \) is ε-close to \( L' \). Thus, there exists an \( M = \max(N_1, N_2) \) (or just \( M = N_1 + N_2 \), if you don't want to use \( \max \)), such that \( a_k \) is ε-close to both \( L \) and \( L' \) for all \( k \ge M \). As we have both \( dist(a_k, L) \le \varepsilon \) and \( dist(a_k, L') \le \varepsilon \), this implies, by the triangle inequality, that \( dist(L, L') \le 2\varepsilon = \frac{2}{3} dist(L, L') \), which is a false statement, as the distance is always positive. Thus, it cannot be true that \( (a_n)_{n=m}^{\infty} \) converges to both \( L \) and \( L' \).

Once it is shown that sequences of reals, if they converge to a real, must converge to a unique real, we can give that unique real a name. It will be called a limit of a sequence (there is a separate card for this definition).

Thus, a limit is simply the real for which a sequence of reals converges to.

Cauchy sequences of reals → convergence of sequences of reals → uniqueness of convergence → limit, the definition → subsumption of formal limits

Source

Tao, Analysis I

Chapter 6