Given a [[Real sequences|sequence]] $(a_{n})_{n=1}^\infty$ we get a subsequence by choosing a sequence of indices $(n_{j})_{j=1}^\infty$, where: $n_{j} \in \mathbb{N} \quad \text{and} \quad n_{j+1}>n_{j} \quad \forall j \in \mathbb{N}$the subsequence is $(a_{n_{j}})_{j=1}^\infty$. # Properties > [!NOTE] Theorem (Any Subsequence of Convergent Sequence Converges to Limit of Sequence) > If $a_{n} \to l$, then any [[Real Subsequence|subsequence]] [[Convergence|converges]] to $l$. >See [[Subsequence of Convergent Sequence Converges to Same Limit|proof]]. # Applications - [[Monotonic Subsequence Theorem]]. - [[If a subsequence of a monotonic sequence converges then the sequence converges]]. - [[Bolzano-Weierstrass Theorem (Sequential Compactness of The Reals)]].