For any complex number `z` prove that `|R e(z)|+|I m(z)|<=sqrt(2)|z|`

For any complex number z prove that |Re(z)|+|Im(z)|<=sqrt(2)|z|

For any complex number z, prove that: (i) -|z| le R_(e)(z)le|z| (ii) -|z|leI_(m)(z)le|z| .

For any two complex numbers z_(1) and prove that quad Re(z_(1)z_(2))=Re z_(1)Re z_(2)-|mz_(1)|mz_(2)

For complex numbers z and w, prove that |z|^(2)w-|w|^(2)z=z-w, if and only if z=w or zbar(w)=1.

For any two complex number z_(1) and z_(2) prove that: |z_(1)+z_(2)|>=|z_(1)|-|z_(2)|

For any two complex number z_(1) and z_(2) prove that: |z_(1)-z_(2)|>=|z_(1)|-|z_(2)|

For any two complex number z_(1) and z_(2) prove that: |z_(1)+z_(2)|<=|z_(1)|+|z_(2)|

For any two complex number z_(1) and z_(2) prove that: |z_(1)-z_(2)|<=|z_(1)|+|z_(2)|

For any two complex numbers z_(1) and z_(2), prove that |z_(1)+z_(2)| =|z_(1)|-|z_(2)| and |z_(1)-z_(2)|>=|z_(1)|-|z_(2)|