Nested radical sequence convergence

How do I prove the sequence $\{\sqrt{7}, \sqrt{7\sqrt{7}},\sqrt{7\sqrt{7\sqrt{7}}}{,... \}}$ converges at 7? I understand intuitively that the final term would be $7^{1/2} \cdot7^{1/4} \cdot7^{1/8}\ldots$ , and that would converge ultimately to $7^1$ but I'm not sure how to properly show that. Thanks!