Let #a# is #real# and #epsilon>0#. Given #V_epsilon(a)={x : |x-a|<epsilon}#. How to find #gamma>0# so that #V_gamma(a)=V_epsilon(a)nnV_delta(a)# ?
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#gamma=min{epsilon, delta}#
If #epsilon < delta #, then #V_(epsilon)(a) cap V_(delta) (a)=V_(epsilon)(a)#, so #gamma=epsilon#.
If #epsilon=delta#, then #V_(epsilon)(a) cap V_(delta) (a)=V_(epsilon)(a)=V_(delta)(a)#, so #gamma=epsilon=delta#.
If #epsilon > delta#, then #V_(epsilon)(a) cap V_(delta) (a)=V_(delta)(a)#, so #gamma=delta#.
Hence, #gamma=min{epsilon, delta}#
I hope that this was clear.
Please see the explanation below for my response to dill's comment.
If #a < b# then
#V_(epsilon)(a)cap V_(delta)(b)=(b-delta,a+epsilon)=V_(gamma)(c)#,
where
#gamma=((a+epsilon)-(b-delta))/2# and #c=((a+epsilon)+(b-delta))/2#.
If #b < a# then
#V_(epsilon)(a)cap V_(delta)(b)=(a-epsilon,b+delta)=V_(gamma)(c)#,
where
#gamma=((b+delta)-(a-epsilon))/2# and #c=((b+delta)+(a-epsilon))/2#.
I hope that this was clear.
