- #1
moriheru
- 273
- 17
I have a question, in the field of number theory (Hardy and Wright chapter 9 representation of numbers by decimals) concerning the prove by contradiction of the statement:
If Σ1∞ an/10n Σ1=∞bn/10n then an and bn must be equivalent, for if not then let aN and bN be the first pair that differ then aN-bN≥1, where an and bn are decimals. It follows that
Σ1∞ an/10n-Σ1∞ bn/10n≥1/10N -ΣN+1∞ (a-b)/10n≥1/10N -∑N+1∞ 9/10n=0
the proof goes on but I only need to know why the sums are greater or smaller than each other...
Thanks for any clarifications, if this is to vague I shall try and add information.
If Σ1∞ an/10n Σ1=∞bn/10n then an and bn must be equivalent, for if not then let aN and bN be the first pair that differ then aN-bN≥1, where an and bn are decimals. It follows that
Σ1∞ an/10n-Σ1∞ bn/10n≥1/10N -ΣN+1∞ (a-b)/10n≥1/10N -∑N+1∞ 9/10n=0
the proof goes on but I only need to know why the sums are greater or smaller than each other...
Thanks for any clarifications, if this is to vague I shall try and add information.