# Thread: A bounded non convergent sequence has two subsequnces with diffrent limits

1. ## A bounded non convergent sequence has two subsequnces with diffrent limits

Show that a bounded non convergent sequence has two sub sequences with different limits.

So using Bolzano- Weierstrass tells us that we have a sub sequence that converges to a real number.

But how do i find two sub sequences with different limits?
I'm guessing we need to construct two of these sub sequences and show that if they're limits are the same it would be a contradiction?

2. Originally Posted by mtlchris
Show that a bounded non convergent sequence has two sub sequences with different limits.

So using Bolzano- Weierstrass tells us that we have a sub sequence that converges to a real number.

But how do i find two sub sequences with different limits?
I'm guessing we need to construct two of these sub sequences and show that if they're limits are the same it would be a contradiction?
So Bolzano–Weierstrass gives you a convergent subsequence, with limit c, say. You also know that the whole sequence does not converge to c.

Now try to write down what it means, in terms of the definition, to say that the sequence does not converge to c. The definition of a sequence converging to c says that given any $\displaystyle \varepsilon>0$, the terms of the sequence eventually get within $\displaystyle \varepsilon$ of c. If that is not true, then there must exist some $\displaystyle \varepsilon>0$ for which infinitely many terms of the sequence are at least distance $\displaystyle \varepsilon$ away from c. These terms have a convergent subsequence (by Bolzano–Weierstrass again), and because each term of that subsequence is at least distance $\displaystyle \varepsilon$ away from c, so is the limit of that subsequence.