CVP3

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Show that |z_{1}+z_{2}|\leq |z_{1}|+|z_{2}|\,.

Let z_{1}=x_{1}+iy_{1},z_{2}=x_{2}+iy_{2}\,.

Show that {\sqrt  {(x_{1}+x_{2})^{2}+(y_{1}+y_{2})^{2}}}\leq {\sqrt  {x_{1}^{2}+y_{1}^{2}}}+{\sqrt  {x_{2}^{2}+y_{2}^{2}}}\,.

Square both sides.

(x_{1}+x_{2})^{2}+(y_{1}+y_{2})^{2}\leq x_{1}^{2}+y_{1}^{2}+2{\sqrt  {(x_{1}^{2}+y_{1}^{2})(x_{2}^{2}+y_{2}^{2})}}+x_{2}^{2}+y_{2}^{2}\,.

2x_{1}x_{2}y_{1}y_{2}\leq x_{1}^{2}y_{2}^{2}+y_{1}^{2}x_{2}^{2}\,

(x_{1}y_{2}-x_{2}y_{1})^{2}\geq 0\,

Complex Variables

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