A question about Tietze extension theroem

Question

Following part of the proof of Tietze extension theorem from "General Topology" by Ryszard Engelking

Well I don't understand why $\sum_{i=1}^{\infty}g_i(x)$ is uniformly convergent to $f(x)$. Could someone help me, please?

Answer 1

Since the series $\displaystyle\sum_{i=1}^\infty\frac12\left(\frac23\right)^{i-1}$ converges and since $\displaystyle(\forall x\in X):\bigl\lvert g_i(x)\bigr\rvert\leqslant\frac12\left(\frac23\right)^{i-1}$, the uniform convergence of your series follows from the Weierstrass $M$-test.

Answer 2

Look up the Weierstrass $M$-test. But it follows immediately from (4), as this holds for all $x$. (Remember that $(2/3)^i\to 0$ as $i\to\infty$.)