Subsequence — Definition, Formula & Examples

A subsequence is a new sequence created by choosing some (or all) terms from an existing sequence while keeping them in their original order. You can skip terms, but you cannot rearrange the ones you keep.

Given a sequence $(a_n)_{n=1}^{\infty}$ , a subsequence is a sequence $(a_{n_k})_{k=1}^{\infty}$ where $n_1 < n_2 < n_3 < \cdots$ is a strictly increasing sequence of positive integers. Each $n_k$ specifies which term of the original sequence is selected as the $k$ -th term of the subsequence.

How It Works

To form a subsequence, pick an infinite collection of indices from the original sequence such that the indices are strictly increasing. The values at those indices, listed in order, give you the subsequence. A key theorem (Bolzano–Weierstrass) states that every bounded sequence of real numbers has a convergent subsequence. Additionally, a sequence converges to

L

if and only if every one of its subsequences also converges to

L

Worked Example

Problem: Consider the sequence

a_n = (-1)^n

for

n = 1, 2, 3, \ldots

, which produces

-1, 1, -1, 1, \ldots

Find a convergent subsequence.

Step 1: Choose the even-indexed terms: set

n_k = 2k

so the indices are

2, 4, 6, 8, \ldots

n_1 = 2,\; n_2 = 4,\; n_3 = 6,\; \ldots

Step 2: Write out the subsequence by evaluating

a_{n_k} = (-1)^{2k}

for each

k

a_{n_k} = (-1)^{2k} = 1 \quad \text{for all } k

Step 3: Since every term of this subsequence equals 1, it converges to 1 — even though the original sequence diverges.

\lim_{k \to \infty} a_{n_k} = 1

Answer: The subsequence

(a_{2k}) = 1, 1, 1, \ldots

converges to

1

Why It Matters

Subsequences are central to real analysis proofs. The Bolzano–Weierstrass theorem — every bounded sequence has a convergent subsequence — is foundational for proving results about continuity, compactness, and the convergence of series. In applied mathematics, extracting a convergent subsequence from approximate solutions is a standard technique for proving existence of exact solutions.

Common Mistakes

Mistake: Thinking a subsequence can reorder terms from the original sequence.

Correction: The indices

n_1 < n_2 < n_3 < \cdots

must be strictly increasing, so the selected terms appear in exactly the same order as in the original sequence.