If $x_n \to a$ and $x'_n \to a$, then $\{x_1, x'_1, x_2, x'_2, ...\} \to a$ I know that if all subsequences of $\{x_1, x'_1, x_2, x'_2, ...\}$ converge to $a$, then $\{x_1, x'_1, x_2, x'_2, ...\}$ converges to $a$, but I only know two subsequences of $\{x_1, x'_1, x_2, x'_2, ...\}$ that converge to $a$, namely $x_n$ and $x'_n$. How do I show that there don't exist any other subsequences that converge to something other than $a$?
 A: Hint: Any convergent sequence is Cauchy. Then notice that every subsequence converging to $b\neq a$ implies that the sequence itself converges to $b$, which generates an absurd.   
A: Fix $\epsilon>0$ and take $N\in\mathbb N$ such that for every $n\ge N$, $|x_n-a|<\epsilon$ and $|x'_n-a|<\epsilon$. Now observe that for this epsilon the natural number $2N$ works.
A: Let $y=(x_1,x_1',x_2,x_2',...)$. For $n$ odd, we have $y_n = x_{n +1\over 2}$,
for $n$ even, we have $y_n = x_{n \over 2}'$.
Let $\epsilon>0$ then there are $N,N'$ such that if
$k \ge N$ we have $|a-x_k| < \epsilon$ and
if $k \ge N'$ we have $|a-x_k'| < \epsilon$.
Let $M = 2\max(N,N')$ and $n \ge M$. Then if $n$ is odd, we have
${n +1\over 2} \ge N$ and so $|y_n-a| < \epsilon$. Similarly, if $n$
is even, we have ${n \over 2}\ge N'$ and so $|y_n-a| < \epsilon$.
Hence $y_n \to a$.
A: HINT:
Use the following lemma:

A sequence $\{a_n\}_n$ converges to $a$ iff every subsequence of $\{a_n\}_n$ admits a sub-subsequence that converges to $a$.

You can try to prove this lemma by your own, it's not difficult (look at the negation of the definition of convergence).
Please notice that the lemma you're trying to use is that a sequence converges to something iff every subsequence of it converges to the same something. The lemma I posted says something different.

Here I show you some details:

 We have to prove that any subsequence of $\{x_1, x'_1, x_2, x'_2, ...\}$ contains a sub-subsequence (i.e., a subsequence of the subsequence) that converges to $a$. Take an arbitrary subsequence of $\{x_1, x'_1, x_2, x'_2, ...\}$, two cases: it contains infinitely many elements from the sequence $\{x_n\}_n$ or it contains infinitely many elements from the sequence $\{x'_n\}_n$, does this give you an idea about how to get a sub-subsequence that converges to $a$?

