# If #a_n# converges and #lim_(n->oo) a_n -b_n=c#, where c is a constant, does #b_n# converge?

Let:

Consider now the quantity:

using the triangular inequality:

which means:

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If ( a_n ) converges and ( \lim_{n \to \infty} (a_n - b_n) = c ), where ( c ) is a constant, then ( b_n ) also converges, and its limit is ( \lim_{n \to \infty} b_n = \lim_{n \to \infty} (a_n - c) ).

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If (a_n) converges and (\lim_{n \to \infty} (a_n - b_n) = c), where (c) is a constant, then (b_n) converges as well.

We know that if (a_n) converges, then (\lim_{n \to \infty} a_n = L) for some finite limit (L).

Given (\lim_{n \to \infty} (a_n - b_n) = c), we can rewrite it as (\lim_{n \to \infty} a_n - \lim_{n \to \infty} b_n = c).

Since (\lim_{n \to \infty} a_n = L) (as (a_n) converges), we have (L - \lim_{n \to \infty} b_n = c).

Rearranging, we get (\lim_{n \to \infty} b_n = L - c), which shows that (b_n) converges to (L - c), a finite limit.

Therefore, if (a_n) converges and (\lim_{n \to \infty} (a_n - b_n) = c), then (b_n) converges.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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