Subsequence

Suppose $u:\mathbb{Z}^+\rightarrow \mathbb{R}$ be a sequence and $v:\mathbb{Z}^+\rightarrow\mathbb{Z}^+$ be an increasing sequence. Then $u\circ v: \mathbb{Z}^+\rightarrow \mathbb{R}$ is a subsequence of $u$.

Monotonic Subsubsequence

Every sequence has a monotonic subsequence.

• Let $n\in\mathbb{Z}^+$ be called “good” iff $\forall m>n,\,u_n > u_m$.
• Suppose $u_n$ has infinitely many “good” points. That implies $u_n$ has a decreasing subsequence.
• Suppose $u_n$ has finitely many “good” points. Let $N$ is the maximum of those. $\forall n_1 > N,\,n_1\;\text{is not "good"}$ That implies $u_n$ has a increasing subsequence.

Bolzano-Weierstrass

Every bounded sequence on $\mathbb{R}$ has a converging subsequence.

From the above theorem, there is a monotonic subsequence $u_{n_k}$ which is also bounded. Bounded monotone sequences converge.

Convergence

Suppose $u_{n_k}$ is a subsequence of $u_n$.

Sequence Converging

Sequence Diverging to Infinity

Converging Subsequence

If $u_n$ is Cauchy and $u_{n_k}$ is a subsequence converging to $L$, then $u_n$ converges to $L$.