Q:

Does taking limits change "$<$" into "$\le$"

I saw this in my notes and didn't think much of it at the time of writing, but now I wanted to make sure this was/wasn't some property of limits no one ever told me about.
Let's say:
$L$ is a limit point of some sequence $S_n$ and each element of the sequence has the property that $s_n \in S_n \to s_n < 1$. Then when you take the limit for "$n$" does this become:
$$
s_n < 1 \Rightarrow \lim_{n\to\infty} s_n \le 1
$$
or does it stay as
$$
s_n < 1 \Rightarrow \lim_{n\to\infty} s_m < 1 \text{ ?}
$$
Thank you!!!

A:

You need $\le$. Take $s_n = 1-\frac{1}{n}$.

A:

The first is correct. There's an for example $s_n=1-2^{-n}$ shows that the second is incorrect.
To see that it's correct you would use reductio ad absurdum. Assum that $\lim s_n=L>1$. Then there should for each $\epsilon>0$ exist an $N$ such that $|s_n-L|<\epsilon$, or $L-\epsilon<s_n<L+\epsilon$ whenever $n>N$. Now select $\epsilon = (L-1)/2>0$, but $s_n < 1$ and $L-\epsilon = L - (L-1)/2 = {L+1}/2 > (1+1)/2 = 1$.

