# Thread: Trouble proving that every convergent sequence is bounded

1. ## Trouble proving that every convergent sequence is bounded

The question and solution for this problem are in problem #5 from the attached pdf. What I don't get at all are the M and |s1|, |s2|, |s3|, ... , |sn0| parts at the end. Also, is the inequality with the equal sign combined technically equal in this case? As for the triangle inequality itself, does the less than part only or usually apply for the magnitudes of vectors or something?

Any help would be greatly appreciated!

2. ## Re: Trouble proving that every convergent sequence is bounded

Originally Posted by s3a
The question and solution for this problem are in problem #5 from the attached pdf. What I don't get at all are the M and |s1|, |s2|, |s3|, ... , |sn0| parts at the end.
If L is the limit of the sequence there exist $\displaystyle n_0$ such that $\displaystyle |s_{s_{n_0}}- L|< 1$. We don't know what the numbers $\displaystyle s_1$, $\displaystyle s_2$, ..., $\displaystyle s_{n_0-1}$ but there are only a finite number, even with L+ 1 added so there is a largest number in that set. That is what M is.

Also, is the inequality with the equal sign combined technically equal in this case?
I'm not sure what yo0u mean by this. Obviously, $\displaystyle |s_n|= |s_n- L+ L|$. By the "triangle inequality", $\displaystyle |(s_n-L)+L\le |s_n-L|+ |L|$ (equality may apply but doesn't have to) and then $\displaystyle |s_n-L|+ |L|$ is strictly less than $\displaystyle 1+ |L|$ because each such $\displaystyle s_n$ was between L-1 and L+1.

]As for the triangle inequality itself, does the less than part only or usually apply for the magnitudes of vectors or something?
The "triangle inequality" applies to positive numbers. The lengths of the sides of a triangle are positive numbers, after all.

Any help would be greatly appreciated!