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

Proof

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

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

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Source

Tao, Analysis I