Is the following a Cauchy sequence? My working:

Fix epsilon>0. I need to show whether an N can be found such that

$|a_m - a_n| \lt \epsilon$ for all $m,n \ge N$

$|a_m - a_n| = |(-1)^m(2+1/m^2) - (-1)^n(2+1/n^2)|$

by taking m even, n odd (is this a good approach?) we get

$|4 + 1/m^2 + 1/n^2| \le |4+2/N^2|$

from here I'm not sure where to go.

Thanks for your help
Simply proove that both sequences have different limits. This is not possible for a Cauchy sequence. 

Why? if $a_{2n}\to l_1$ and $a_{2n+1}\to l_2$m take $$\epsilon = \frac{|l_1 - l_2|}3$$

There is a $N$ such as $\forall n\ge N \  |a_{2n} - l_1| + |a_{2n+1} - l_2| &lt; \epsilon$. 

Then, for instance, $|a_{2n} - a_{2N+1}| \ge |l_1 - l_2| - (|a_{2n} - l_1| + |a_{2n+1} - l_2|)\ge \epsilon $.

Here, $a_{2n}\to 2$ and $a_{2n+1}\to -2$.
Short answer, no.  To verify, evaluate the limit of this sequence