combine convegence question..

**transgalactic** · December 4th 2008, 01:32 PM

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

??

**Mathstud28** · December 4th 2008, 01:37 PM

Originally Posted by transgalactic

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

??

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

??

**Plato** · December 4th 2008, 01:54 PM

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

**Mathstud28** · December 4th 2008, 02:07 PM

Originally Posted by Plato

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

Did I screw up in my proof? Yours looks exactly the same?

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