Subsequences

We have seen some bounded sequences which do not converge. We can, however, say something about such sequences.

Definition

A subsequence is an infinite ordered subset of a sequence.

Examples

(a₂ , a₄ , a₆ , ... ) is a subsequence of (a₁ , a₂ , a₃ , a₄ , ... ). So is (a₁ , a₁₀ , a₁₀₀ , a₁₀₀₀ , ... ).

Theorem

Any subsequence of a convergent sequence is convergent (to the same limit).

Proof

Look at the definition!

The nicest thing about these subsequences is a result attributed to the Czech mathematician and philosopher Bernard Bolzano (1781 to 1848) and the German mathematician Karl Weierstrass (1815 to 1897).

The Bolzano-Weierstrass Theorem

Every bounded sequence has a convergent subsequence.

Remark

Notice that a bounded sequence may have many convergent subsequences (for example, a sequence consisting of a counting of the rationals has subsequences converging to every real number) or rather few (for example a convergent sequence has all its subsequences having the same limit).

Proof

Suppose the sequence (a₁ , a₂ , a₃ , a₄ , ... ) is bounded and lies in (say) the interval [0, 1].
Then we construct a convergent subsequence by a bisection process.
Split the interval [0, 1] into two halves [0 , ¹/₂] and [¹/₂ , 1]. Then (at least) one of the halves will contain infinitely many terms of the sequence. Suppose it is [0 , ¹/₂]. Choose x₁to be one of these terms.
Then split this interval in half again and repeat the process choosing x₂ to be further down the sequence than x₁. Continuing in this way, at the nth stage we will choose a term x_n lying in an interval [a_n , b_n].

Claim: The subsequence (x_n) is convergent.

Proof of claim
The sequence (l_n) of "Left-hand ends" of intervals is monotonic increasing, bounded above by 1 and hence has a limit α.
The sequence (r_n) of "right-hand ends" of intervals is monotonic decreasing, bounded below by 0 and hence has a limit β.
Since the length of the interval [l_n , r_n] has length (¹/₂)ⁿ, we must have α = β and since the sequence (x_n) is trapped between (l_n) and (r_n), it converges to the same limit.