So, by an informal proof I can show this is true: i. e., by creating a connected planar graph with $6$ nodes with one node of degree $5$, and $5$ nodes of degree $1$. Then by adding edges to a node with currently degree $1$, I can increase that node to degree $5$, eventually there will be a point where I can no longer add any edges without breaking the condition of planarity. And will have remaining nodes with degree no more than $5$.
Is there a better way to approach this? Perhaps with Euler's Theorem which states that a planar graph will satisfy the following condition: $V - E + R = 2$, where $v$ is vertices; $e$ is edges; $r$ is regions and we know that sum of all vertex degrees is equivalent to $2E$.