Does $\lim_{n\to\infty}S_n \leq S$ imply that $S_n < S$?

Question

Does $\lim_{n\to\infty}S_n \leq S$ imply that $S_n < S$ ?

Some something from proof I am working out bothering me.

only when $S_n$ is decreasing/converging – RE60K Sep 19 '14 at 05:06 — RE60K, Sep 19 '14 at 05:06
I think you meant increasing, @Aditya – Timbuc Sep 19 '14 at 05:07 — Timbuc, Sep 19 '14 at 05:07
ok thank yes, he meant increasing i think too – Pasie15 Sep 19 '14 at 05:11 — Pasie15, Sep 19 '14 at 05:11

score 3 · Answer 1 · answered Sep 19 '14 at 05:07

3

If $S_n$ is an arbitrary sequence, then this is not the case. For example, take the sequence $$ \{1,0,1,1/2,1,3/4,1,7/8,1,\dots\} $$

answered Sep 19 '14 at 05:07

Ben Grossmann

225,327

Your example doesn't seem to converge, contrary to the OP's assumption. – Timbuc Sep 19 '14 at 05:07
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@Timbuc this example converges to $1$. – Ben Grossmann Sep 19 '14 at 05:08
ok...but then all the elements of your sequence are less than or equal its limit. – Timbuc Sep 19 '14 at 05:10
2

@Timbuc which contradicts the given statement, the limit is $\le 1$ but there are terms which $=1$. – DanZimm Sep 19 '14 at 05:12
so if the sequence is increaibg this statement is right, correct ? – Pasie15 Sep 19 '14 at 05:13
Indeed so, @DanZimm. Thanks, I must have missed that stron inequality sign. – Timbuc Sep 19 '14 at 05:13
@Timbuc all good - I do the same thing all the time! – DanZimm Sep 19 '14 at 05:19

Kim Jong Un · Answer 2 · 2014-09-19T05:13:15.600

3

Let $S_n=1+\frac{(-1)^n}{n}$. Then $S_n\to 1\leq 1$ but $S_n$ with even $n$ will be greater than $1$. In fact, $\{S_n\}$ will alternate between above and below $1$.

edited Sep 19 '14 at 05:13

answered Sep 19 '14 at 05:07

Kim Jong Un

14,794
1
24
49

score 2 · Answer 3 · answered Sep 19 '14 at 05:12

2

For a very simple counter example take

$$\frac{n+1}n\xrightarrow[n\to\infty]{}1\;,\;\;\text{yet}\;\;\;\frac{n+1}n>1\;\;\;\;\forall\;n\in\Bbb N$$

answered Sep 19 '14 at 05:12

Timbuc

34,191

Does $\lim_{n\to\infty}S_n \leq S$ imply that $S_n < S$?

3 Answers3