A **subsequential limit **is the limit of a *subsequence*; a **subsequence** is a smaller part of some larger sequence. For example, the integer sequence {1, 2, 1, 2} has two subsequences: one for odd numbers and one for even numbers.

Sometimes a limit doesn’t exist for a sequence, but *does* exist for one or more subsequences. For example, the limit for the following sequence doesn’t exist:

However, there are three subsequences with limits [1]:

- (x
_{1}, x_{4}, x_{7}, …) converges to 1. - (x
_{2}, 5_{1}, 8_{1}, …) converges to 0. - (x
_{3}, x_{6}, x_{9}, …) converges to -1.

## Examples: Finding a Subsequential Limit

**Example question #1:** Find the subsequential limits of the sequence a_{n} = 1 – (-1)^{n}.

The key here is to look for a pattern by generating some terms.

Step 1: **Generate some terms of the sequence**. One of the easiest ways to do this is in Excel. Here’s the steps:

- Type the numbers 1 through 10 in column A of a worksheet.
- Type the formula in cell B2. When you get to the “n”, click on cell A1.
- Press Enter, then grab the little square at the right hand corner of B2. Drag this square to the bottom of column B2.

Step 2: **Identify the pattern.** We have two subsequences here: odd numbers where the limit is 2 and even numbers where the limit is 0. Therefore, the subsequential limits are 0 and 2.

**Example question #2:** Find the subsequential limits of the sequence a_{n} = 1 – (-1)^{n}.

Step 1: **Generate some terms of the sequence**. For this example, I didn’t see a pattern for the first ten terms, so I went much larger:

Step 2: **Identify the pattern.**The sequence (and all of its subsequences) converge to 1. This is an example where it may be easier to demonstrate the limit with algebra. We know that |a_{n}| = 1/n → -, so a_{n} → 1.

The limits of subsequences can more formally be defined by the limit inferior and limit superior.

## References

Upper and Lower Limits of Sequences of Real Numbers. https://faculty.math.illinois.edu/~r-ash/RV/RV3.pdf

**CITE THIS AS:**

**Stephanie Glen**. "Subsequential Limit" From

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