If `z_1a n dz_2` are two complex numbers and `c >0` , then prove that `|z_1+z_2|^2lt=(1+c)|z_1|^2+(1+c^(-1))|z_2|^2dot`

Explanation

We have to prove
`|z_(1)+z_(2)|^(2)le(1+c)|z_(1)|^(2)+(1+c^(-1))|z_(2)|^(2)`
or `|z_(1)|^(2)+|z_(2)|^(2)+z_(1)bar(z)_(2)le(1+c)|z_(1)|^(2)+(1+c^(-1))|z_(2)|^(2)`
or `z_(1)bar(z)_(2)+bar(z)_(1)z_(2)lec|z_(1)|^(2)+c^(-1)|z_(2)|^(2)`
or `c|z_(1)|^(2)+(1)/(c)|z_(2)|^(2)-z_(1)bar(z)_(2)-bar(z)_(1)z_(2)ge0" "["using Re "(z_(1)bar(z)_(2))le|z_(1)bar(z)_(2)|]`
or `|sqrt(c)z_(1)-(1)/(sqrt(c))z_(2)|^(2)ge0`
which is always true.