# Thread: combine convegence question..

1. ## combine convegence question..

prove that if An->a and An-Bn->0 then Bn->a

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2. Originally Posted by transgalactic
prove that if An->a and An-Bn->0 then Bn->a

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http://www.mathhelpforum.com/math-he...-question.html

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3. If $\displaystyle \varepsilon > 0$ then $\displaystyle \left( {\exists N} \right)\left[ {n \geqslant N \Rightarrow \left| {A_n - a} \right| < \frac{\varepsilon }{2}\,\& \,\left| {A_n - B_n } \right| < \frac{\varepsilon }{2}} \right]$.
So $\displaystyle \left| {B_n - a} \right| \leqslant \left| {B_n - A_n } \right| + \left| {A_n - a} \right| < \varepsilon$.

4. Originally Posted by Plato
If $\displaystyle \varepsilon > 0$ then $\displaystyle \left( {\exists N} \right)\left[ {n \geqslant N \Rightarrow \left| {A_n - a} \right| < \frac{\varepsilon }{2}\,\& \,\left| {A_n - B_n } \right| < \frac{\varepsilon }{2}} \right]$.
So $\displaystyle \left| {B_n - a} \right| \leqslant \left| {B_n - A_n } \right| + \left| {A_n - a} \right| < \varepsilon$.
Did I screw up in my proof? Yours looks exactly the same?