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Let $x_n$ be a sequence. If $M =\limsup x_n$ then there is some subsequence $x_k$ with $M = \lim x_k$. Then $$ \left|x_k - M \right|< \epsilon \iff M - \epsilon < x_k < M+\epsilon \quad \forall k>K, $$

Then how are we saying that $x_k>M-\epsilon$ for infinitely many $k$ but $x_k < M+\epsilon$ for $k>K$?

Is my concept about the limit sup not correct? If so please help me with the correct concept about limit sup of a sequence.

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You can have that $x_k > M-\epsilon$ and $x_k < M+\epsilon$ at the same time. For example, let $M=2$, $\epsilon=0.1$ and $x_k = 1.99$. These values satisfy the inequality $2-0.1 < 1.99 < 2+0.1$. When $\epsilon$ starts getting smaller and smaller, you just need $x_k$ to get closer and closer to $M$ (this may require that $k$ increases too).

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Not sure what your question is exactly. In your case, there is an interval $(M-\epsilon,M+\epsilon)$, where $\epsilon$ depends on the specific $K$, such that if you take any $k>K$, then all members of the subsequence $\left\{x_{n_k}\right\}$ will fall inside this interval.

It is expected that as $K \to \infty$ you will have $\epsilon \to 0$.

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