Cauchy-Convergence Criteria Ques : Show by Cauchy-Convergence criteria that sequence  given by an = 1+1/3+1/5+..1/2n-1
is not convergent.
My proof: Let is assume it is convergnt then it would be cauchy (By cauchy convergence criteria). Then for $\epsilon$=1/4 there exist m $\varepsilon$ Z such that |an-am|<$\epsilon $ for all n>m.
Let n=2m
|an-am|=|1/2m + 1/2m+1 +....+1/2n-1|>|1/2m + ....+ 1/2n|    (1)
=|1/2m +......+ 1/4m|>|1/4m +..1/4m |=m/4m=1/4 so we have proved for n=2m >m
|an-am|>=1/4 so contradictio0n.
so it is not convergent .
IS it correct and can we do (1).. plz tell if there is different way

Your proof is correct with a minor fix at the term $\dfrac{1}{2m}$ to be deleted. Following your work up to: $|a_{2m} - a_m| = |\dfrac{1}{2m+1} + \dfrac{1}{2m+3} +...+ \dfrac{1}{4m-1}|$. As you counted already that there are $m$ terms in the absolute value, and you replaced each denominator by $4m$, so $|a_{2m} - a_m| > \dfrac{m}{4m} = \dfrac{1}{4}$.
