Limits and subsucesions If you are working with a monotonous function, it is possible to calculate a limit using a subsucesion, but if you get the same limit independently the subsucesion you use, can I claim that it is the limit of the function?

If a function has a limit $$\lim_{x\to\infty}f(x)=a,$$ then for every sequence $x_n\to\infty$ it is true that $$f(x_n)\to a.$$ It is not true the other way around (choose $f(x)=\sin(x)$ and $x_n=\pi n$).

If a sequence converges, then so does every subsequence. If a subsequence converges, then the sequence does not converge in general (choose $a_n=(-1)^n$ and the subsequence $a_{2n}$).