proving that a sequence is basis

Question

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Q1 : In proof of part a) in proposition 8.1 from book "Ivan singer, Bases in Banach spaces I"
I don't know why he write that $$\displaystyle\sup_{1 \leq n < \infty} \|\sum_{i=1}^n \alpha_i x_i\| \leq \lim_{n \rightarrow \infty} \displaystyle\sup_{1 \leq k \leq n} \|\displaystyle\sum_{i=1}^k \alpha_i x_i\|$$ where $A_1$ is the Banach space $\{ \{\alpha_n\} \subset K : \sum_{i=1}^\infty \alpha_i x_i \text{ converges }\}$ endowed with the norm $\|\{\alpha_n\}\| = \displaystyle\sup_{1 \leq n \leq \infty} \|\sum_{i=1}^n \alpha_i x_i\|$ Thanks in advance