Why does the existence of distinct limits for the odd and even subsequences of $a_n=(-1)^n$ prove the sequence diverges?

A fundamental theorem states that if a sequence converges to

L

, every subsequence must also converge to the same

L

. Here, odd terms approach

-1

and even terms approach

1

. Since

-1 \neq 1

, the sequence cannot converge to any single value.

Conditions

Theorem: Convergence implies unique subsequential limits.
Odd subsequence limit is $-1$ .
Even subsequence limit is $1$ .

Reasoning, step by step

Identify the odd subsequence: $a_{2k+1} = -1$ . Limit is $-1$ .
Identify the even subsequence: $a_{2k} = 1$ . Limit is $1$ .
Compare the two limits: $-1 \neq 1$ .
Apply the necessary condition for convergence: All subsequences must share the same limit.
Conclude divergence because the condition is violated.

Example

The summary explicitly states: 'Equivalently, a convergent sequence must give its odd and even subsequences the same limit, but theirs are −1 and 1.' This provides an intuitive shortcut alongside the rigorous epsilon proof.

Common misconceptions

Believing that boundedness implies convergence.
Thinking that oscillating between two values is a form of convergence to the average.

Watch the explanation

BilibiliDisproving a sequence limit from the definition

1:18 – 1:34Watch this moment ↗

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Why does the existence of distinct limits for the odd and even subsequences of an=(−1)na_n=(-1)^nan​=(−1)n prove the sequence diverges?